{"id":32344,"date":"2021-05-09T13:00:24","date_gmt":"2021-05-09T13:00:24","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=32344"},"modified":"2025-03-03T22:38:59","modified_gmt":"2025-03-03T22:38:59","slug":"%e0%a4%85%e0%a4%96%e0%a4%82%e0%a4%a1%e0%a4%a4%e0%a4%be-%e0%a4%a1%e0%a5%8b%e0%a4%ae%e0%a5%87%e0%a4%a8-%e0%a4%94%e0%a4%b0-%e0%a4%aa%e0%a5%82%e0%a4%b0%e0%a5%8d%e0%a4%a3%e0%a4%be%e0%a4%82%e0%a4%95","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/hi\/%e0%a4%85%e0%a4%96%e0%a4%82%e0%a4%a1%e0%a4%a4%e0%a4%be-%e0%a4%a1%e0%a5%8b%e0%a4%ae%e0%a5%87%e0%a4%a8-%e0%a4%94%e0%a4%b0-%e0%a4%aa%e0%a5%82%e0%a4%b0%e0%a5%8d%e0%a4%a3%e0%a4%be%e0%a4%82%e0%a4%95\/","title":{"rendered":"\u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0914\u0930 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901"},"content":{"rendered":"<style>\n\tp, ul, ol{\n\ttext-align: justify;\n\t}\n\th1{\n\ttext-align:center;\n\ttext-transform: uppercase;\n\t}\n\th2{\n\ttext-align:center;\n\ttext-transform: uppercase;\n\tfont-size:24pt;\n\t}\n\th3 { \n\t\ttext-align: center;\n\t\ttext-transform: uppercase;\n\t\tfont-size: 24px !important;\n\t}\n\t<\/style>\n<h1>\u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0914\u0930 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901<\/h1>\n<p style=\"text-align:center;\"><em><strong>\u0938\u093e\u0930\u093e\u0902\u0936:<\/strong><br \/>\n\t\u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u092e\u0947\u0902 \u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0915\u0940 \u0905\u0935\u0927\u093e\u0930\u0923\u093e \u0915\u093e \u092a\u0930\u093f\u091a\u092f \u0926\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948, \u0907\u0938\u0915\u0947 \u092e\u0939\u0924\u094d\u0935 \u0915\u094b \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0915\u0947 \u0905\u0927\u094d\u092f\u092f\u0928 \u092e\u0947\u0902 \u0938\u092e\u091d\u093e\u092f\u093e \u0917\u092f\u093e \u0939\u0948, \u0914\u0930 \u0907\u0938\u0915\u0940 \u0915\u0941\u091b \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u092a\u094d\u0930\u092e\u093e\u0923\u094b\u0902 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948\u0964 <\/em><\/p>\n<p style=\"text-align:center;\"><em><strong>\u0938\u0940\u0916\u0928\u0947 \u0915\u0947 \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f:<\/strong><\/em><br \/>\n\t\u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u0915\u0947 \u0905\u0902\u0924 \u0924\u0915, \u091b\u093e\u0924\u094d\u0930 \u0938\u0915\u094d\u0937\u092e \u0939\u094b\u0902\u0917\u0947:\n\t<\/p>\n<ol>\n<li><strong>\u0938\u092e\u091d\u0947\u0902<\/strong> \u0915\u093f \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0915\u093e \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u094d\u092f\u094b\u0902 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 <\/li>\n<li><strong>\u0938\u092e\u091d\u0947\u0902<\/strong> \u0915\u093f \u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0915\u0940 \u0905\u0935\u0927\u093e\u0930\u0923\u093e \u0915\u094d\u092f\u093e \u0939\u0948\u0964 <\/li>\n<li><strong>\u0938\u092e\u091d\u093e\u090f\u0901<\/strong> \u0915\u093f \u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0914\u0930 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u092c\u0940\u091a \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u092e\u094c\u0932\u093f\u0915 \u0935\u093f\u0936\u0947\u0937\u0924\u093e\u090f\u0901 \u0915\u094d\u092f\u093e \u0939\u0948\u0902\u0964 <\/li>\n<li><strong>\u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902<\/strong> \u0915\u093f \u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0915\u0947 \u092e\u0942\u0932\u092d\u0942\u0924 \u0917\u0941\u0923 \u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u092a\u094d\u0930\u092e\u093e\u0923\u094b\u0902 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u0938\u0924\u094d\u092f\u093e\u092a\u093f\u0924 \u0915\u093f\u090f \u091c\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964 <\/li>\n<\/ol>\n<p style=\"text-align:center;\">\n\t<strong><u>\u0938\u093e\u092e\u0917\u094d\u0930\u0940 \u0938\u0942\u091a\u0940<\/u><\/strong><br \/>\n\t<a href=\"#1\">\u0938\u093e\u092e\u093e\u0928\u094d\u092f \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0915\u093e \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f \u0914\u0930 \u092a\u0942\u0930\u094d\u0935 \u091c\u094d\u091e\u093e\u0928<\/a><br \/>\n\t<a href=\"#2\">\u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0938\u0947 \u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0924\u0915<\/a><br \/>\n\t<a href=\"#3\">\u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0914\u0930 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u092e\u094c\u0932\u093f\u0915 \u092a\u0939\u0932\u0942<\/a><br \/>\n\t<a href=\"#4\">\u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0914\u0930 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0917\u0941\u0923<\/a><br \/>\n\t<a href=\"#5\">\u0905\u092d\u094d\u092f\u093e\u0938<\/a>\n\t<\/p>\n<p>\t<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/hxmc1-eXWxU?si=57GADT52JG4fHFT-\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><br \/>\n\t<\/center><br \/>\n\t<a name=\"1\"><\/a><\/p>\n<h2>\u0938\u093e\u092e\u093e\u0928\u094d\u092f \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0915\u093e \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f \u0914\u0930 \u092a\u0942\u0930\u094d\u0935 \u091c\u094d\u091e\u093e\u0928<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=hxmc1-eXWxU&amp;t=183s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0938\u093e\u092e\u093e\u0928\u094d\u092f \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0915\u093e \u092e\u0941\u0916\u094d\u092f \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f<\/span><\/strong><\/a> \u0938\u092d\u0940 \u0938\u0902\u092d\u093e\u0935\u093f\u0924 \u0917\u0923\u093f\u0924\u0940\u092f \u092a\u094d\u0930\u0923\u093e\u0932\u093f\u092f\u094b\u0902 \u0915\u093e \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u0930\u0928\u093e \u0939\u0948\u0964 \u092f\u0939\u093e\u0902, \u0939\u092e \u0915\u0908 \u0910\u0938\u0940 \u092a\u094d\u0930\u0923\u093e\u0932\u093f\u092f\u094b\u0902 \u0915\u093e \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u0930\u0947\u0902\u0917\u0947, \u0914\u0930 \u0909\u0928\u092e\u0947\u0902 \u0938\u0947 \u0938\u092c\u0938\u0947 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0914\u0930 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0939\u0948\u0902\u0964 \u0907\u0928\u094d\u0939\u0940\u0902 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u0939\u092e \u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0924\u0915 \u092a\u0939\u0941\u0901\u091a\u0947\u0902\u0917\u0947\u0964<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{N}= \\{1,2,3,4,\\cdots\\}<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{Z}= \\{0,\\pm 1,\\pm 2,\\pm 3,\\pm 4,\\cdots\\}<\/span><\/span><\/p>\n<p>\t<a name=\"2\"><\/a><\/p>\n<h2>\u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0938\u0947 \u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0924\u0915<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=hxmc1-eXWxU&amp;t=358s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0939\u092e \u0905\u092a\u0928\u0947 \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u0940 \u0936\u0941\u0930\u0941\u0906\u0924 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0938\u0947 \u0915\u0930\u0947\u0902\u0917\u0947,<\/span><\/strong><\/a> \u0914\u0930 \u0910\u0938\u093e \u0915\u0930\u0928\u0947 \u0915\u093e \u0915\u093e\u0930\u0923 \u092f\u0939 \u0939\u0948 \u0915\u093f \u0935\u0947 \u0909\u0928 \u0905\u0927\u093f\u0915\u093e\u0902\u0936 \u0938\u0902\u0916\u094d\u092f\u093e\u0924\u094d\u092e\u0915 \u092a\u094d\u0930\u0923\u093e\u0932\u093f\u092f\u094b\u0902 \u0938\u0947 \u0938\u092c\u0938\u0947 \u0905\u0927\u093f\u0915 \u0938\u092e\u093e\u0928\u0924\u093e \u0930\u0916\u0924\u0947 \u0939\u0948\u0902 \u091c\u093f\u0928\u094d\u0939\u0947\u0902 \u0939\u092e \u0907\u0938 \u0905\u0927\u094d\u092f\u092f\u0928 \u092e\u0947\u0902 \u0926\u0947\u0916\u0947\u0902\u0917\u0947\u0964<\/p>\n<p>\u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0915\u094d\u092f\u093e \u0939\u0948\u0902, \u0907\u0938\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0928\u0947 \u0915\u093e \u092a\u094d\u0930\u092f\u093e\u0938 \u0915\u0930\u0928\u0947 \u0915\u0947 \u092c\u091c\u093e\u092f, \u0939\u092e \u092f\u0939 \u092e\u093e\u0928\u0915\u0930 \u0936\u0941\u0930\u0941\u0906\u0924 \u0915\u0930\u0947\u0902\u0917\u0947 \u0915\u093f \u0935\u0947 \u091c\u094b \u092d\u0940 \u0939\u094b\u0902, \u0915\u0941\u091b \u0928\u093f\u0936\u094d\u091a\u093f\u0924 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964 \u0907\u0938\u0915\u0947 \u0932\u093f\u090f, \u0939\u092e \u0910\u0938\u0947 \u0938\u094d\u0935\u092f\u0902\u0938\u093f\u0926\u094d\u0927 (axioms) \u091a\u0941\u0928\u0924\u0947 \u0939\u0948\u0902 \u091c\u094b \u0939\u092e\u0947\u0902 \u0909\u0928 \u0938\u092d\u0940 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u0928\u093f\u0915\u093e\u0932\u0928\u0947 \u0915\u0940 \u0905\u0928\u0941\u092e\u0924\u093f \u0926\u0947\u0924\u0947 \u0939\u0948\u0902 \u091c\u093f\u0928\u094d\u0939\u0947\u0902 \u0939\u092e \u0938\u0939\u091c \u0930\u0942\u092a \u0938\u0947 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0938\u0947 \u091c\u094b\u0921\u093c\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n<p>\u092f\u0947 \u0938\u092d\u0940 \u091a\u0940\u091c\u0947\u0902 <strong>\u092a\u0940\u092f\u093e\u0928\u094b \u0938\u094d\u0935\u092f\u0902\u0938\u093f\u0926\u094d\u0927<\/strong> (Peano Axioms) \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948\u0902, \u091c\u0939\u093e\u0901 \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u092a\u0930 \u0905\u0902\u0915\u0917\u0923\u093f\u0924\u0940\u092f \u0938\u0902\u0915\u094d\u0930\u093f\u092f\u093e\u090f\u0901 (operations) \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948\u0902\u0964 \u0907\u0938 \u0938\u094d\u0935\u092f\u0902\u0938\u093f\u0926\u094d\u0927 \u092a\u0926\u094d\u0927\u0924\u093f \u0915\u093e \u0905\u0928\u0941\u0938\u0930\u0923 \u0915\u0930\u0915\u0947 \u0914\u0930 \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u090f\u0935\u0902 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u092a\u0930 \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u0938\u0902\u0915\u094d\u0930\u093f\u092f\u093e\u0913\u0902 \u0915\u093e \u0935\u093f\u0938\u094d\u0924\u093e\u0930 \u0915\u0930\u0915\u0947, \u0939\u092e\u0947\u0902 \u0928\u090f \u0938\u0902\u0916\u094d\u092f\u093e\u0924\u094d\u092e\u0915 \u0938\u092e\u0942\u0939 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902, \u091c\u0948\u0938\u0947 \u0915\u093f \u092a\u0930\u093f\u092e\u0947\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 (rational numbers), \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 (irrational numbers), \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 (real numbers), \u0938\u092e\u094d\u092e\u093f\u0936\u094d\u0930 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 (complex numbers), \u0915\u094d\u0935\u093e\u091f\u0930\u0928\u093f\u092f\u0928 (quaternions), \u0911\u0915\u094d\u091f\u094b\u0928\u093f\u092f\u0928 (octonions) \u0914\u0930 \u0907\u0938\u0940 \u092a\u094d\u0930\u0915\u093e\u0930 \u0905\u0928\u0947\u0915 \u0905\u0928\u094d\u092f\u0964<\/p>\n<p>\u0907\u0938\u0915\u0947 \u092c\u093e\u0926, \u092f\u0926\u093f \u0939\u092e \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u0926\u0947\u0916\u0947\u0902, \u0924\u094b \u092a\u093e\u090f\u0902\u0917\u0947 \u0915\u093f \u0907\u0928\u092e\u0947\u0902 \u0915\u0941\u091b \u0917\u0941\u0923 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902 \u091c\u094b \u0905\u0928\u094d\u092f \u0938\u092d\u0940 \u0938\u0902\u0916\u094d\u092f\u093e\u0924\u094d\u092e\u0915 \u0938\u092e\u0942\u0939\u094b\u0902 \u092e\u0947\u0902 \u0926\u094b\u0939\u0930\u093e\u090f \u091c\u093e\u0924\u0947 \u0939\u0948\u0902, \u091c\u0948\u0938\u0947 \u0915\u093f \u0917\u0941\u0923\u0928 \u0915\u0947 \u0932\u093f\u090f \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 (multiplicative identity), \u092f\u094b\u0917 \u0915\u0947 \u0932\u093f\u090f \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 (additive identity), \u0914\u0930 \u0935\u093f\u0924\u0930\u0923\u0940\u092f \u0928\u093f\u092f\u092e (distributive laws)\u0964 \u0907\u0928 \u0905\u0935\u0927\u093e\u0930\u0923\u093e\u0913\u0902 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947, \u0939\u092e \u090f\u0915 \u0910\u0938\u0940 \u092d\u093e\u0937\u093e \u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u091c\u094b \u0939\u092e\u0947\u0902 \u0907\u0928 \u0938\u092d\u0940 \u0938\u092e\u0942\u0939\u094b\u0902 \u0915\u0947 \u092c\u093e\u0930\u0947 \u092e\u0947\u0902 \u090f\u0915 \u0938\u093e\u0925 \u091a\u0930\u094d\u091a\u093e \u0915\u0930\u0928\u0947 \u0915\u0940 \u0905\u0928\u0941\u092e\u0924\u093f \u0926\u0947\u0924\u0940 \u0939\u0948\u0964 \u0907\u0938\u0940 \u0938\u0902\u0926\u0930\u094d\u092d \u092e\u0947\u0902, \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0936\u092c\u094d\u0926 \u0909\u092d\u0930\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<ul>\n<li>\u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 (Integrity Domain)<\/li>\n<li>\u0935\u0932\u092f (Ring)<\/li>\n<li>\u0938\u092e\u0942\u0939 (Group)<\/li>\n<li>\u0938\u0926\u093f\u0936\u0940\u092f \u0938\u094d\u0925\u093e\u0928 (Vector Space)<\/li>\n<\/ul>\n<p>\u0914\u0930 \u0907\u0938\u0940 \u0924\u0930\u0939 \u0915\u0908 \u0905\u0928\u094d\u092f \u0917\u0923\u093f\u0924\u0940\u092f \u0936\u092c\u094d\u0926&#8230; \u0939\u092e \u0905\u092a\u0928\u0940 \u090a\u0930\u094d\u091c\u093e \u092a\u0939\u0932\u0947 <strong>\u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928<\/strong> \u0915\u093e \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u0930\u0928\u0947 \u092e\u0947\u0902 \u0932\u0917\u093e\u090f\u0902\u0917\u0947\u0964<\/p>\n<p>\t<a name=\"3\"><\/a><\/p>\n<h3>\u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0914\u0930 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u092e\u094c\u0932\u093f\u0915 \u092a\u0939\u0932\u0942<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=hxmc1-eXWxU&amp;t=472s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0915\u094d\u092f\u093e \u0939\u0948<\/span><\/strong><\/a> \u0907\u0938\u0947 \u0938\u092e\u091d\u093e\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f, \u0939\u092e \u0909\u0928 \u0917\u0941\u0923\u094b\u0902 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0947\u0902\u0917\u0947 \u091c\u093f\u0928\u094d\u0939\u0947\u0902 \u0939\u092e \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0938\u0947 \u0905\u091a\u094d\u091b\u0940 \u0924\u0930\u0939 \u0938\u092e\u091d\u0924\u0947 \u0939\u0948\u0902\u0964 \u0907\u0938 \u0938\u0902\u0926\u0930\u094d\u092d \u092e\u0947\u0902, \u092f\u0926\u093f <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0939\u0948\u0902, \u0924\u094b \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 <strong>\u0928\u093f\u092f\u092e<\/strong> \u0938\u0924\u094d\u092f \u0939\u094b\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<ol>\n<li><strong>\u092a\u0930\u093f\u0935\u0930\u094d\u0924\u0928\u0940\u092f\u0924\u093e (Commutativity):<\/strong>\n<ul>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+b = b + a<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ab = ba<\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<li><strong>\u0938\u0939\u0938\u0902\u092f\u094b\u091c\u0915\u0924\u093e (Associativity):<\/strong>\n<ul>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+(b+c) = a+b+c = (a+b)+c<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(ab)c = abc = a(bc)<\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<li><strong>\u0935\u093f\u0924\u0930\u0923\u0940\u092f\u0924\u093e (Distributivity):<\/strong>\n<ul>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+(b+c) = a(b+c) = ab+ac<\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p>\u0907\u0938\u0915\u0947 \u0905\u0932\u093e\u0935\u093e, \u0915\u0941\u091b \u0935\u093f\u0936\u0947\u0937 \u0924\u0924\u094d\u0935 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902 \u091c\u093f\u0928\u094d\u0939\u0947\u0902 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 (neutral elements) \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/p>\n<ul>\n<li><strong>\u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 (Additive Identity):<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+ c = a \\leftrightarrow c=0<\/span><\/span><\/li>\n<li><strong>\u0917\u0941\u0923\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 (Multiplicative Identity):<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ac = a \\leftrightarrow c=1<\/span><\/span><\/li>\n<\/ul>\n<p>\u091c\u093f\u0938 \u0935\u0938\u094d\u0924\u0941 \u0915\u093e \u092a\u094d\u0930\u0924\u0940\u0915 <span class=\"katex-eq\" data-katex-display=\"false\">0<\/span> \u0939\u0948, \u0909\u0938\u0947 \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u0914\u0930 \u091c\u093f\u0938\u0915\u093e \u092a\u094d\u0930\u0924\u0940\u0915 <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> \u0939\u0948, \u0935\u0939 \u0917\u0941\u0923\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/p>\n<p>\u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u092a\u093e\u0938 \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e (additive inverses) \u092d\u0940 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u0947 \u0932\u093f\u090f \u090f\u0915 \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e \u0939\u094b\u0924\u093e \u0939\u0948, \u091c\u094b \u0909\u0938\u0938\u0947 \u091c\u094b\u0921\u093c\u0928\u0947 \u092a\u0930 \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 \u092a\u094d\u0930\u0926\u093e\u0928 \u0915\u0930\u0924\u093e \u0939\u0948\u0964<\/p>\n<ul>\n<li><strong>\u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e (Additive Inverse):<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+ c = 0 \\longleftrightarrow c=-a<\/span><\/span><\/li>\n<\/ul>\n<p>\u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e \u0915\u094b \u0909\u0928\u0915\u0947 \u0938\u093e\u0925 \u091c\u0941\u0921\u093c\u0947 \u091a\u093f\u0939\u094d\u0928 \u00ab-\u00bb \u0938\u0947 \u092a\u0939\u091a\u093e\u0928\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/p>\n<p>\u0905\u0902\u0924\u0924\u0903, \u090f\u0915 <strong>\u0938\u0930\u0932\u0940\u0915\u0930\u0923 \u0928\u093f\u092f\u092e (Law of Simplification)<\/strong> \u0939\u094b\u0924\u093e \u0939\u0948, \u091c\u093f\u0938\u0947 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0902\u092c\u0902\u0927 \u0926\u094d\u0935\u093e\u0930\u093e \u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(c\\neq 0 \\wedge ca = cb) \\longleftrightarrow (a=b)<\/span><\/span><\/p>\n<p>\u092f\u0947 \u0917\u0941\u0923 \u091c\u094b \u0939\u092e\u0928\u0947 \u0926\u0947\u0916\u0947 \u0939\u0948\u0902, \u0915\u0908 \u0905\u0928\u094d\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u0924\u094d\u092e\u0915 \u0938\u092e\u0942\u0939\u094b\u0902 \u091c\u0948\u0938\u0947 \u0915\u093f \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901, \u0938\u092e\u094d\u092e\u093f\u0936\u094d\u0930 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901, \u092c\u0939\u0941\u092a\u0926 (polynomials) \u0906\u0926\u093f \u092a\u0930 \u092d\u0940 \u0932\u093e\u0917\u0942 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964 \u0907\u0938\u0932\u093f\u090f, \u0909\u0928 \u0938\u092d\u0940 \u0938\u092e\u0942\u0939\u094b\u0902 \u0915\u094b \u091c\u094b \u0907\u0928 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0915\u0930\u0924\u0947 \u0939\u0948\u0902, <strong>\u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 (Integrity Domain)<\/strong> \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/p>\n<p><span style=\"color: #800000;\"><strong>\u092a\u0930\u093f\u092d\u093e\u0937\u093e (Definition):<\/strong><\/span> \u090f\u0915 \u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0935\u0939 \u0915\u094b\u0908 \u092d\u0940 \u0938\u092e\u0942\u0939 <span class=\"katex-eq\" data-katex-display=\"false\">D<\/span> \u0939\u094b\u0924\u093e \u0939\u0948, \u091c\u093f\u0938\u092e\u0947\u0902 \u091c\u094b\u0921\u093c\u0928\u0947 \u0914\u0930 \u0917\u0941\u0923\u093e \u0915\u0930\u0928\u0947 \u0915\u0940 \u0938\u0902\u0915\u094d\u0930\u093f\u092f\u093e\u090f\u0901 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u094b\u0924\u0940 \u0939\u0948\u0902 \u0915\u093f:<\/p>\n<ul>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in D \\longrightarrow a+b \\in D<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in D \\longrightarrow ab \\in D<\/span><\/span><\/li>\n<\/ul>\n<p>\u0907\u0938\u0915\u0947 \u0905\u0932\u093e\u0935\u093e, <strong>\u0938\u0939\u0938\u0902\u092f\u094b\u091c\u0915\u0924\u093e (associativity), \u092a\u0930\u093f\u0935\u0930\u094d\u0924\u0928\u0940\u092f\u0924\u093e (commutativity)<\/strong> \u0914\u0930 <strong>\u0935\u093f\u0924\u0930\u0923\u0940\u092f\u0924\u093e (distributivity)<\/strong> \u0915\u0947 \u0928\u093f\u092f\u092e \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964 <span class=\"katex-eq\" data-katex-display=\"false\">D<\/span> \u092e\u0947\u0902 <strong>\u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 (additive identity)<\/strong> \u0914\u0930 <strong>\u0917\u0941\u0923\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 (multiplicative identity)<\/strong> \u0939\u094b\u0924\u0947 \u0939\u0948\u0902 (\u091c\u094b \u0905\u0926\u094d\u0935\u093f\u0924\u0940\u092f \u0939\u094b\u0924\u0947 \u0939\u0948\u0902), \u0914\u0930 \u0905\u0902\u0924\u0924\u0903, <strong>\u0938\u0930\u0932\u0940\u0915\u0930\u0923 \u0928\u093f\u092f\u092e (law of simplification)<\/strong> \u092d\u0940 \u0932\u093e\u0917\u0942 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/p>\n<h4>\u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0915\u093e \u0909\u0926\u093e\u0939\u0930\u0923<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=hxmc1-eXWxU&amp;t=749s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0939\u092e \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u092e\u0942\u0939 \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0930\u0947\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A=\\{a+b\\sqrt{3}\\; |\\; a,b\\in \\mathbb{Z}\\}.<\/span><\/span><\/span> <\/strong><\/a> \u092f\u0939 \u0938\u092e\u0942\u0939, \u091c\u092c \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u091c\u094b\u0921\u093c \u0914\u0930 \u0917\u0941\u0923\u093e \u0938\u0902\u0915\u094d\u0930\u093f\u092f\u093e\u0913\u0902 \u0915\u0947 \u0938\u093e\u0925 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u0924\u094b \u090f\u0915 \u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u092c\u0928\u093e\u0924\u093e \u0939\u0948, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u092f\u0939 \u092a\u0930\u093f\u0935\u0930\u094d\u0924\u0928\u0940\u092f\u0924\u093e, \u0938\u0939\u0938\u0902\u092f\u094b\u091c\u0915\u0924\u093e \u0914\u0930 \u0935\u093f\u0924\u0930\u0923\u0940\u092f\u0924\u093e \u0915\u0947 \u0928\u093f\u092f\u092e\u094b\u0902 \u0915\u094b \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0915\u0930\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u092e\u0947\u0902 \u090f\u0915 \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935, \u090f\u0915 \u0917\u0941\u0923\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 \u0914\u0930 \u090f\u0915 \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/p>\n<ul>\n<li><strong>\u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0+0\\sqrt{3}<\/span><\/span><\/li>\n<li><strong>\u0917\u0941\u0923\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1+0\\sqrt{3}<\/span><\/span><\/li>\n<li><strong>\u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e:<\/strong> \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0924\u0924\u094d\u0935 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+b\\sqrt{3}<\/span><\/span> \u0915\u093e \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-a-b\\sqrt{3}<\/span><\/span> \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/li>\n<\/ul>\n<p>\u0914\u0930 \u0938\u092c\u0938\u0947 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u092c\u093e\u0924, \u092f\u0939 \u0938\u092e\u0942\u0939 A \u091c\u094b\u0921\u093c \u0914\u0930 \u0917\u0941\u0923\u093e \u0938\u0902\u0915\u094d\u0930\u093f\u092f\u093e\u0913\u0902 \u0915\u0947 \u0932\u093f\u090f \u092c\u0902\u0926 (closed) \u0939\u0948, \u091c\u093f\u0938\u0915\u093e \u0905\u0930\u094d\u0925 \u0939\u0948 \u0915\u093f \u092f\u0926\u093f \u0939\u092e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x,y\\in A<\/span><\/span> \u0932\u0947\u0902, \u0924\u094b \u092f\u0939 \u0938\u0924\u094d\u092f \u0939\u094b\u0917\u093e \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x+y\\in A<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">xy\\in A.<\/span><\/span> \u0907\u0938\u0947 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u093e \u0938\u0930\u0932 \u0939\u0948: \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a_1 + b_1\\sqrt{3}<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a_2 + b_2\\sqrt{3}<\/span><\/span> \u0924\u0924\u094d\u0935 <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u0915\u0947 \u0938\u0926\u0938\u094d\u092f \u0939\u0948\u0902, \u0924\u094b:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\t(a_1 + b_1\\sqrt{3}) + (a_2 + b_2\\sqrt{3}) &amp;=(a_1+a_2) + (b_1 + b_2)\\sqrt{3} \\in A\\\\ \\\\\n\n\t(a_1 + b_1\\sqrt{3})  (a_2 + b_2\\sqrt{3})  &amp;= a_1a_2 + a_1b_2\\sqrt{3}+b_1a_2\\sqrt{3} + 3b_1b_2 \\\\\n\n\t&amp;=(a_1a_2 + 3b_1b_2) + (a_1b_2 + b_1a_2)\\sqrt{3} \\in A\n\n\t\\end{array}<\/span>\n<p>\t<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/y6jXNPhjKv4?si=9SaXhWHN42sC73lZ\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<p>\t<a name=\"4\"><\/a><\/p>\n<h3>\u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0914\u0930 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0917\u0941\u0923<\/h3>\n<h4>\u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u092e\u0947\u0902 \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 \u0905\u0926\u094d\u0935\u093f\u0924\u0940\u092f \u0939\u094b\u0924\u093e \u0939\u0948<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=26s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0907\u0938\u0947 \u0935\u093f\u0938\u0902\u0917\u0924\u093f \u0926\u094d\u0935\u093e\u0930\u093e \u0905\u092a\u093e\u0928\u092f\u0928 (reductio ad absurdum) \u0938\u0947 \u0938\u093f\u0926\u094d\u0927 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948:<\/span><\/strong><\/a> \u092e\u093e\u0928 \u0932\u0947\u0902 \u0915\u093f \u0926\u094b \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 \u092e\u094c\u091c\u0942\u0926 \u0939\u0948\u0902, \u0905\u0930\u094d\u0925\u093e\u0924\u094d <span class=\"katex-eq\" data-katex-display=\"false\">0<\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0^\\prime<\/span><\/span>\u0964 \u0924\u092c \u0939\u092e\u0947\u0902 \u092e\u093f\u0932\u0947\u0917\u093e:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n\t(1) &amp; 0\\neq 0^\\prime &amp; \\text{; \u092a\u094d\u0930\u0924\u094d\u092f\u092f (Premise)}\\\\\n\n\t(2) &amp; a+0 = a &amp; \\text{; \u092a\u094d\u0930\u0924\u094d\u092f\u092f: $0$ \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 \u0939\u0948}\\\\\n\n\t(3) &amp; b+0^\\prime = b &amp; \\text{; \u092a\u094d\u0930\u0924\u094d\u092f\u092f: $0^\\prime$ \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 \u0939\u0948}\\\\\n\n\t(4) &amp; 0^\\prime + 0 = 0^\\prime &amp; \\text{; $(2)$ \u092e\u0947\u0902 $a=0^\\prime$ \u092a\u094d\u0930\u0924\u093f\u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u0930\u0915\u0947}\\\\\n\n\t(5) &amp; 0 + 0^\\prime = 0 &amp; \\text{; $(3)$ \u092e\u0947\u0902 $b=0$ \u092a\u094d\u0930\u0924\u093f\u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u0930\u0915\u0947}\\\\\n\n\t(6) &amp; 0 = 0^\\prime  &amp; \\text{; $(4,5)$ \u0914\u0930 \u092f\u094b\u0917 \u0915\u0940 \u092a\u0930\u093f\u0935\u0930\u094d\u0924\u0928\u0936\u0940\u0932\u0924\u093e (commutativity) \u0938\u0947}\\\\\n\n\t(7) &amp; \\bot &amp;\\text{; $(1,6)$ \u0938\u0947 \u0935\u093f\u0938\u0902\u0917\u0924\u093f \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u0940 \u0939\u0948}\n\n\t\\end{array}<\/span>\n<p>\u0907\u0938 \u0924\u0930\u094d\u0915 \u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930, \u0939\u092e \u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937 \u0928\u093f\u0915\u093e\u0932\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{0 \\neq 0^\\prime, a + 0 = a, b + 0^\\prime = b\\}\\vdash \\bot.<\/span><\/span><\/p>\n<p>\u0907\u0938\u0932\u093f\u090f, \u0935\u093f\u0938\u0902\u0917\u0924\u093f \u0926\u094d\u0935\u093e\u0930\u093e \u0905\u092a\u093e\u0928\u092f\u0928 \u0938\u0947, \u0939\u092e\u0947\u0902 \u092e\u093f\u0932\u0924\u093e \u0939\u0948<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{a + 0 = a, b + 0^\\prime = b\\}\\vdash 0 = 0^\\prime.<\/span><\/span><\/p>\n<p>\u0905\u0930\u094d\u0925\u093e\u0924, \u092f\u0926\u093f \u0926\u094b \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 \u092e\u094c\u091c\u0942\u0926 \u0939\u094b\u0902, \u0924\u094b \u0935\u0947 \u0938\u092e\u093e\u0928 \u0939\u094b\u0928\u0947 \u091a\u093e\u0939\u093f\u090f, \u0914\u0930 \u0907\u0938\u0932\u093f\u090f \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 \u0905\u0926\u094d\u0935\u093f\u0924\u0940\u092f \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/p>\n<h4>\u0917\u0941\u0923\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 \u092d\u0940 \u0905\u0926\u094d\u0935\u093f\u0924\u0940\u092f \u0939\u094b\u0924\u093e \u0939\u0948<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=305s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0907\u0938\u0915\u093e \u092a\u094d\u0930\u092e\u093e\u0923 \u092d\u0940 \u0932\u0917\u092d\u0917 \u092a\u093f\u091b\u0932\u0947 \u092a\u094d\u0930\u092e\u093e\u0923 \u0915\u0947 \u0938\u092e\u093e\u0928 \u0939\u0948\u0964<\/span><\/strong><\/a> \u092f\u0926\u093f \u0926\u094b \u0917\u0941\u0923\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1^\\prime<\/span><\/span> \u092e\u094c\u091c\u0942\u0926 \u0939\u094b\u0902, \u0924\u094b \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0924\u0930\u094d\u0915 \u092a\u094d\u0930\u0938\u094d\u0924\u0941\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n\t(1) &amp; 1\\neq 1^\\prime &amp; \\text{; \u092a\u094d\u0930\u0924\u094d\u092f\u092f (Premise)}\\\\\n\n\t(2) &amp; 1\\cdot a = a &amp; \\text{; \u092a\u094d\u0930\u0924\u094d\u092f\u092f: $1$ \u0917\u0941\u0923\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 \u0939\u0948}\\\\\n\n\t(3) &amp; 1^\\prime \\cdot b = b &amp; \\text{; \u092a\u094d\u0930\u0924\u094d\u092f\u092f: $1^\\prime$ \u0917\u0941\u0923\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 \u0939\u0948}\\\\\n\n\t(4) &amp; 1\\cdot 1^\\prime = 1^\\prime &amp; \\text{; $(2)$ \u092e\u0947\u0902 $a=1^\\prime$ \u092a\u094d\u0930\u0924\u093f\u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u0930\u0915\u0947}\\\\\n\n\t(5) &amp; 1^\\prime \\cdot 1 = 1 &amp; \\text{; $(3)$ \u092e\u0947\u0902 $b=1$ \u092a\u094d\u0930\u0924\u093f\u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u0930\u0915\u0947}\\\\\n\n\t(6) &amp; 1 = 1^\\prime  &amp; \\text{; $(4,5)$ \u0914\u0930 \u0917\u0941\u0923\u093e \u0915\u0940 \u092a\u0930\u093f\u0935\u0930\u094d\u0924\u0928\u0936\u0940\u0932\u0924\u093e (commutativity) \u0938\u0947}\\\\\n\n\t(7) &amp; \\bot &amp;\\text{; $(1,6)$ \u0938\u0947 \u0935\u093f\u0938\u0902\u0917\u0924\u093f \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u0940 \u0939\u0948}\n\n\t\\end{array}<\/span>\n<p>\u0907\u0938\u0932\u093f\u090f, \u0939\u092e \u0907\u0938 \u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937 \u092a\u0930 \u092a\u0939\u0941\u0901\u091a\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{1 \\neq 1^\\prime, 1a= a, 1b = b\\}\\vdash \\bot.<\/span><\/span><\/p>\n<p>\u0905\u0924\u0903, \u0935\u093f\u0938\u0902\u0917\u0924\u093f \u0926\u094d\u0935\u093e\u0930\u093e \u0905\u092a\u093e\u0928\u092f\u0928 (reductio ad absurdum) \u0938\u0947, \u0939\u092e\u0947\u0902 \u092e\u093f\u0932\u0924\u093e \u0939\u0948:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{1a= a, 1b= b\\}\\vdash 1 = 1^\\prime.<\/span><\/span><\/p>\n<p>\u0905\u0930\u094d\u0925\u093e\u0924, \u092f\u0926\u093f \u0926\u094b \u0917\u0941\u0923\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 \u092e\u094c\u091c\u0942\u0926 \u0939\u094b\u0902, \u0924\u094b \u0935\u0947 \u0938\u092e\u093e\u0928 \u0939\u094b\u0928\u0947 \u091a\u093e\u0939\u093f\u090f, \u0914\u0930 \u0907\u0938\u0932\u093f\u090f \u092f\u0939 \u0905\u0926\u094d\u0935\u093f\u0924\u0940\u092f \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/p>\n<h4>\u092f\u094b\u0917 (\u0938\u0902\u0915\u0932\u0928) \u0915\u0947 \u0932\u093f\u090f \u0938\u0930\u0932\u0940\u0915\u0930\u0923 \u0915\u093e \u0928\u093f\u092f\u092e \u092e\u093e\u0928\u094d\u092f \u0939\u0948<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=461s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0939\u092e \u092f\u0939\u0940 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902 \u091c\u092c<\/span><\/strong><\/a> \u0939\u092e \u0915\u093f\u0938\u0940 \u0938\u092e\u093e\u0928\u0924\u093e (equality) \u092e\u0947\u0902 \u0938\u0947 \u092a\u0926\u094b\u0902 \u0915\u094b \u0939\u091f\u093e\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+b = a+c \\longleftrightarrow a = c<\/span><\/span><\/p>\n<p>\u0907\u0938 \u0938\u094d\u0925\u093f\u0924\u093f \u0915\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u093e \u0915\u0920\u093f\u0928 \u0928\u0939\u0940\u0902 \u0939\u0948, \u0915\u0947\u0935\u0932 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0924\u0930\u094d\u0915 \u0915\u094b \u0905\u092a\u0928\u093e\u0928\u0947 \u0915\u0940 \u0906\u0935\u0936\u094d\u092f\u0915\u0924\u093e \u0939\u0948:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n\t(1) &amp; a+b = a+c &amp; \\text{; \u092a\u094d\u0930\u0924\u094d\u092f\u092f (Premise)} \\\\\n\n\t(2) &amp; a+b-a = a+c-a &amp; \\text{; $(1)$ \u0938\u0947, \u0926\u094b\u0928\u094b\u0902 \u092a\u0915\u094d\u0937\u094b\u0902 \u092e\u0947\u0902 $-a$ \u091c\u094b\u0921\u093c\u0915\u0930} \\\\\n\n\t(3) &amp; (a-a)+b = (a-a)+c &amp; \\text{; $(2)$ \u0938\u0947, \u092a\u0930\u093f\u0935\u0930\u094d\u0924\u0928\u0936\u0940\u0932\u0924\u093e (commutativity) \u0914\u0930 \u0938\u0939\u0938\u0902\u092f\u094b\u091c\u0915\u0924\u093e (associativity)} \\\\\n\n\t(4) &amp; 0+b = 0+c &amp; \\text{; $(3)$ \u0914\u0930 \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e (additive inverse) \u0938\u0947} \\\\\n\n\t(5) &amp; b = c &amp; \\text{; $(4)$ \u0914\u0930 \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 (additive identity) \u0938\u0947} \\\\\n\n\t\\end{array}<\/span>\n<p>\u091a\u0942\u0901\u0915\u093f \u092f\u0939 \u0924\u0930\u094d\u0915 \u0906\u0917\u0947 \u0914\u0930 \u092a\u0940\u091b\u0947 \u0926\u094b\u0928\u094b\u0902 \u0926\u093f\u0936\u093e\u0913\u0902 \u092e\u0947\u0902 \u0938\u092e\u093e\u0928 \u091a\u0930\u0923\u094b\u0902 \u0938\u0947 \u0932\u093e\u0917\u0942 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948, \u0907\u0938\u0932\u093f\u090f<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+b=a+c \\dashv \\vdash b=c<\/span><\/span><\/p>\n<p>\u091c\u094b \u0907\u0938\u0915\u0947 \u0938\u092e\u0924\u0941\u0932\u094d\u092f \u0939\u0948:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash a+b=a+c \\longleftrightarrow b=c<\/span><\/span><\/p>\n<h4>\u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935, \u0917\u0941\u0923\u093e \u0915\u0947 \u0932\u093f\u090f \u090f\u0915 \u0905\u0935\u0936\u094b\u0937\u0915 (absorbent) \u0924\u0924\u094d\u0935 \u092d\u0940 \u0939\u0948<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=632s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0907\u0938\u0915\u093e \u0938\u0940\u0927\u093e \u0938\u093e \u092e\u0924\u0932\u092c \u092f\u0939 \u0939\u0948<\/span><\/strong><\/a> \u0915\u093f \u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u092e\u0947\u0902 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> \u0915\u0947 \u0932\u093f\u090f, \u092f\u0939 \u0938\u0924\u094d\u092f \u0939\u094b\u0917\u093e \u0915\u093f:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\cdot 0 = 0<\/span><\/span><\/p>\n<p>\u0907\u0938\u0915\u093e \u092a\u094d\u0930\u092e\u093e\u0923 \u092d\u0940 \u0938\u0930\u0932 \u0939\u0948, \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0924\u0930\u094d\u0915 \u0915\u094b \u0926\u0947\u0916\u0947\u0902:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n\t(1) &amp; a\\cdot a + a\\cdot 0 = a\\cdot  (a+0) &amp; \\text{; \u0935\u093f\u0924\u0930\u0923\u0940\u092f\u0924\u093e (Distributive laws)}\\\\\n\n\t(2) &amp; a\\cdot a + a\\cdot 0 = a\\cdot  (a+a-a) &amp; \\text{; $(1)$ \u0914\u0930 \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e \u0938\u0947} \\\\\n\n\t(3) &amp; a\\cdot a + a\\cdot 0 = a\\cdot a + a\\cdot a - a\\cdot a &amp; \\text{; $(2)$ \u0914\u0930 \u0935\u093f\u0924\u0930\u0923\u0940\u092f\u0924\u093e \u0938\u0947} \\\\\n\n\t(4) &amp;  a\\cdot 0 =  a\\cdot a - a\\cdot a &amp; \\text{; $(3)$ \u0914\u0930 \u0938\u0902\u0915\u0932\u0928 \u0915\u0947 \u0938\u0930\u0932\u0940\u0915\u0930\u0923 (simplification) \u0938\u0947} \\\\\n\n\t(5) &amp;  a\\cdot 0 =  0 &amp; \\text{; $(4)$ \u0914\u0930 \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e \u0938\u0947} \\\\\n\n\t\\end{array}<\/span>\n<h4><strong>\u091a\u093f\u0939\u094d\u0928\u094b\u0902 \u0915\u093e \u0928\u093f\u092f\u092e (Law of Signs):<\/strong><\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=736s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0938\u092e\u093e\u0928 \u091a\u093f\u0939\u094d\u0928 \u0935\u093e\u0932\u0940 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0917\u0941\u0923\u0928\u092b\u0932<\/span><\/strong><\/a> \u0939\u092e\u0947\u0936\u093e \u0927\u0928\u093e\u0924\u094d\u092e\u0915 (positive) \u0939\u094b\u0924\u093e \u0939\u0948, \u091c\u092c\u0915\u093f \u0935\u093f\u092a\u0930\u0940\u0924 \u091a\u093f\u0939\u094d\u0928\u094b\u0902 \u0935\u093e\u0932\u0940 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0917\u0941\u0923\u0928\u092b\u0932 \u0939\u092e\u0947\u0936\u093e \u090b\u0923\u093e\u0924\u094d\u092e\u0915 (negative) \u0939\u094b\u0924\u093e \u0939\u0948\u0964 \u0907\u0938 \u0917\u0941\u0923\u0927\u0930\u094d\u092e \u0915\u093e \u092a\u094d\u0930\u092e\u093e\u0923 \u092d\u0940 \u0938\u0930\u0932 \u0939\u0948:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n\t(1) &amp; a\\cdot b = a\\cdot b + 0 &amp; \\text{; \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935 (Additive Identity)}\\\\\n\n\t(2) &amp; a\\cdot b = a\\cdot b + (a)\\cdot(-b) - (a)\\cdot(-b) &amp; \\text{; $(1)$ \u0914\u0930 \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e (Additive Inverse)}\\\\\n\n\t(3) &amp; a\\cdot b = a\\cdot (b -b) - (a)\\cdot(-b) &amp; \\text{; $(2)$ \u0914\u0930 \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e \u0938\u0947} \\\\\n\n\t(4) &amp; a\\cdot b = a\\cdot 0 + (-a)\\cdot(-b) &amp; \\text{; $(3)$ \u0914\u0930 \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e \u0938\u0947} \\\\\n\n\t(5) &amp; a\\cdot b = (-a)\\cdot(-b) &amp; \\text{; $(4)$ \u0914\u0930 \u0917\u0941\u0923\u093e\u0924\u094d\u092e\u0915 \u0905\u0935\u0936\u094b\u0937\u0915 (Multiplicative Absorption) \u0938\u0947} \\\\\n\n\t\\end{array}<\/span>\n<p>\u0905\u0924\u0903: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ab = (-a)(-b)<\/span><\/span><\/p>\n<p>\u0935\u093f\u092a\u0930\u0940\u0924 \u091a\u093f\u0939\u094d\u0928\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u092d\u0940 \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e \u0938\u092e\u093e\u0928 \u0939\u094b\u0924\u0940 \u0939\u0948:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n\t(1) &amp; a\\cdot(-b) = a \\cdot (-b) + 0 &amp; \\text{; \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0924\u091f\u0938\u094d\u0925 \u0924\u0924\u094d\u0935} \\\\\n\n\t(2) &amp; a\\cdot(-b) = a \\cdot (-b) + a \\cdot b  -  a \\cdot b  &amp; \\text{; $(1)$ \u0914\u0930 \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e \u0938\u0947} \\\\\n\n\t(3) &amp; a\\cdot(-b) = a \\cdot (b-b)  -  a \\cdot b  &amp; \\text{; $(2)$ \u0914\u0930 \u0935\u093f\u0924\u0930\u0923\u0940\u092f\u0924\u093e (Distributivity) \u0938\u0947} \\\\\n\n\t(4) &amp; a\\cdot(-b) = a \\cdot 0  -  a \\cdot b  &amp; \\text{; $(3)$ \u0914\u0930 \u0938\u0902\u0935\u0930\u094d\u0927\u0928\u093e\u0924\u094d\u092e\u0915 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e \u0938\u0947} \\\\\n\n\t(5) &amp; a\\cdot(-b) = - a \\cdot b  &amp; \\text{; $(4)$ \u0914\u0930 \u0917\u0941\u0923\u093e\u0924\u094d\u092e\u0915 \u0905\u0935\u0936\u094b\u0937\u0915 \u0938\u0947} \\\\\n\n\t\\end{array}<\/span>\n<p>\u0905\u0924\u0903: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a(-b) = -a(b)<\/span><\/span><\/p>\n<h4>\u092f\u0926\u093f \u0926\u094b \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0917\u0941\u0923\u0928\u092b\u0932 \u0936\u0942\u0928\u094d\u092f \u0939\u0948, \u0924\u094b \u0915\u092e \u0938\u0947 \u0915\u092e \u090f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e \u0936\u0942\u0928\u094d\u092f \u0939\u094b\u0917\u0940<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=875s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u090f\u0915 \u0914\u0930 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0917\u0941\u0923\u0927\u0930\u094d\u092e<\/span><\/strong><\/a> \u091c\u094b \u0905\u0915\u094d\u0938\u0930 \u0909\u092a\u092f\u094b\u0917 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u0935\u0939 \u092f\u0939 \u0939\u0948:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ab=0 \\leftrightarrow (a=0 \\vee b=0)<\/span><\/span><\/p>\n<p>\u0907\u0938\u0915\u093e \u092a\u094d\u0930\u092e\u093e\u0923 \u092d\u0940 \u0938\u0930\u0932 \u0939\u0948:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n\t(1) &amp; \\{a=0\\} \\models a\\cdot b = 0  &amp; \\textbf{; \u0917\u0941\u0923\u093e\u0924\u094d\u092e\u0915 \u0905\u0935\u0936\u094b\u0937\u0915 (Multiplicative Absorption)} \\\\\n\n\t(2) &amp; \\models a=0 \\rightarrow a\\cdot b = 0  &amp;\\text{; TD$(1)$} \\\\\n\n\t(3) &amp; \\models \\neg (a\\cdot b = 0 ) \\rightarrow \\neg(a=0) &amp;\\text{; CPI$(2)$} \\\\\n\n\t(4) &amp; \\{\\neg (a\\cdot b = 0 ) \\}\\models   \\neg(a=0) &amp;\\text{; RTD$(3)$} \\\\\n\n\t(5) &amp; \\{\\neg (a\\cdot b = 0 ) \\}\\models   \\neg(b=0) &amp;\\text{; \u0905\u0928\u0941\u0930\u0942\u092a $(4)$} \\\\\n\n\t(6) &amp; \\{\\neg (a\\cdot b = 0 ) \\}\\models  \\neg(a=0) \\wedge \\neg(b=0) &amp;\\text{; $\\wedge$-int$(4,5)$} \\\\\n\n\t(7) &amp; \\models (\\neg (a\\cdot b = 0 )) \\rightarrow \\neg(a=0) \\wedge \\neg(b=0)  &amp;\\text{; TD(6)} \\\\\n\n\t(8) &amp; \\models \\neg(\\neg(a=0) \\wedge \\neg(b=0) ) \\rightarrow   (a\\cdot b = 0 ) &amp;\\text{; CPI(7)} \\\\\n\n\t(9) &amp; \\models (a=0 \\vee b=0) \\rightarrow   (a\\cdot b = 0 ) &amp;\\text{; DM(8)} \\\\\n\n\t(10)&amp; \\{a\\neq 0 , a\\cdot b=0\\} \\models b=0 &amp; \\textbf{; \u0917\u0941\u0923\u093e\u0924\u094d\u092e\u0915 \u0905\u0935\u0936\u094b\u0937\u0915 (Multiplicative Absorption)} \\\\\n\n\t(11)&amp; \\{a\\cdot b=0\\} \\models a\\neq 0 \\rightarrow  b=0 &amp; \\text{; TD(10)} \\\\\n\n\t(12)&amp; \\{a\\cdot b=0\\} \\models \\neg(a\\neq 0) \\vee  b=0 &amp; \\text{; $\\rightarrow$-Def(11)} \\\\\n\n\t(13)&amp; \\{a\\cdot b=0\\} \\models a=0 \\vee  b=0 &amp; \\text{; DN(12)} \\\\\n\n\t(14)&amp; \\models (a\\cdot b=0) \\rightarrow (a=0 \\vee  b=0) &amp; \\text{; TD(13)} \\\\\n\n\t(15)&amp; \\models (a\\cdot b=0) \\leftrightarrow (a=0 \\vee  b=0) &amp; \\text{; De(9,14)}\n\n\t\\end{array}<\/span>\n<p>\t<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/KZQ3PXeMlKk?si=O_Hek5KFG853Q6qT\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<p>\t<a name=\"5\"><\/a><\/p>\n<h2>\u0905\u092d\u094d\u092f\u093e\u0938<\/h2>\n<p>\u092e\u093e\u0928 \u0932\u0947\u0902 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> \u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 <span class=\"katex-eq\" data-katex-display=\"false\">D<\/span> \u0915\u0947 \u0915\u094b\u0908 \u092d\u0940 \u0924\u0924\u094d\u0935 \u0939\u0948\u0902\u0964 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0917\u0941\u0923\u0927\u0930\u094d\u092e \u0938\u0924\u094d\u092f \u0939\u0948\u0902:<\/p>\n<ol>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(-a)=(-1)a<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=306s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[\u0938\u092e\u093e\u0927\u093e\u0928]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-(a+b)=(-a) + (-b)<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=827s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[\u0938\u092e\u093e\u0927\u093e\u0928]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a(-b)=-(ab)<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=1213s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[\u0938\u092e\u093e\u0927\u093e\u0928]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-(-a)=a<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=1628s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[\u0938\u092e\u093e\u0927\u093e\u0928]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a(b-c) = ab - ac<\/span><\/span> <strong>[\u092a\u094d\u0930\u0938\u094d\u0924\u093e\u0935\u093f\u0924]<\/strong><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a-b)+(b-c) = a-c<\/span><\/span> <strong>[\u092a\u094d\u0930\u0938\u094d\u0924\u093e\u0935\u093f\u0924]<\/strong><\/li>\n<li>\u0938\u092d\u0940 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in D<\/span><\/span> \u0915\u0947 \u0932\u093f\u090f, \u090f\u0915 \u0905\u0926\u094d\u0935\u093f\u0924\u0940\u092f <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> \u092e\u094c\u091c\u0942\u0926 \u0939\u094b\u0924\u093e \u0939\u0948, \u0910\u0938\u093e \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\cdot 1 = a<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=2029s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[\u0938\u092e\u093e\u0927\u093e\u0928]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">xx = x \\leftrightarrow (x=1 \\vee x=0)<\/span><\/span> <strong>[\u092a\u094d\u0930\u0938\u094d\u0924\u093e\u0935\u093f\u0924]<\/strong><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0914\u0930 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0938\u093e\u0930\u093e\u0902\u0936: \u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u092e\u0947\u0902 \u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0915\u0940 \u0905\u0935\u0927\u093e\u0930\u0923\u093e \u0915\u093e \u092a\u0930\u093f\u091a\u092f \u0926\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948, \u0907\u0938\u0915\u0947 \u092e\u0939\u0924\u094d\u0935 \u0915\u094b \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0915\u0947 \u0905\u0927\u094d\u092f\u092f\u0928 \u092e\u0947\u0902 \u0938\u092e\u091d\u093e\u092f\u093e \u0917\u092f\u093e \u0939\u0948, \u0914\u0930 \u0907\u0938\u0915\u0940 \u0915\u0941\u091b \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u092a\u094d\u0930\u092e\u093e\u0923\u094b\u0902 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948\u0964 \u0938\u0940\u0916\u0928\u0947 \u0915\u0947 \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f: \u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u0915\u0947 \u0905\u0902\u0924 \u0924\u0915, \u091b\u093e\u0924\u094d\u0930 \u0938\u0915\u094d\u0937\u092e \u0939\u094b\u0902\u0917\u0947: [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":32317,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":2,"footnotes":""},"categories":[577,1039],"tags":[],"class_list":["post-32344","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-577","category-1039"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>\u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928 \u0914\u0930 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"\u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u092e\u0947\u0902 \u0905\u0916\u0902\u0921\u0924\u093e \u0921\u094b\u092e\u0947\u0928, \u0909\u0928\u0915\u0947 \u0917\u0941\u0923\u0927\u0930\u094d\u092e \u0914\u0930 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