{"id":26669,"date":"2021-03-29T13:00:34","date_gmt":"2021-03-29T13:00:34","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=26669"},"modified":"2024-05-21T10:07:34","modified_gmt":"2024-05-21T10:07:34","slug":"%e0%a4%aa%e0%a5%8d%e0%a4%b0%e0%a4%be%e0%a4%af%e0%a4%bf%e0%a4%95%e0%a4%a4%e0%a4%be-%e0%a4%97%e0%a4%a3%e0%a4%a8%e0%a4%be-%e0%a4%95%e0%a5%87-%e0%a4%b2%e0%a4%bf%e0%a4%8f-%e0%a4%89%e0%a4%aa%e0%a4%af","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/hi\/%e0%a4%aa%e0%a5%8d%e0%a4%b0%e0%a4%be%e0%a4%af%e0%a4%bf%e0%a4%95%e0%a4%a4%e0%a4%be-%e0%a4%97%e0%a4%a3%e0%a4%a8%e0%a4%be-%e0%a4%95%e0%a5%87-%e0%a4%b2%e0%a4%bf%e0%a4%8f-%e0%a4%89%e0%a4%aa%e0%a4%af\/","title":{"rendered":"\u092a\u094d\u0930\u093e\u092f\u093f\u0915\u0924\u093e \u0917\u0923\u0928\u093e \u0915\u0947 \u0932\u093f\u090f \u0909\u092a\u092f\u094b\u0917\u0940 \u092a\u094d\u0930\u092e\u0947\u092f"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>\u092a\u094d\u0930\u093e\u092f\u093f\u0915\u0924\u093e \u0917\u0923\u0928\u093e \u0915\u0947 \u0932\u093f\u090f \u0909\u092a\u092f\u094b\u0917\u0940 \u092a\u094d\u0930\u092e\u0947\u092f<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>\u0938\u093e\u0930\u093e\u0902\u0936<\/strong><br \/><em>\u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u092e\u0947\u0902, \u0915\u0941\u091b \u092a\u094d\u0930\u093e\u092f\u093f\u0915\u0924\u093e \u0917\u0923\u0928\u093e \u0915\u0947 \u0932\u093f\u090f \u0909\u092a\u092f\u094b\u0917\u0940 \u092a\u094d\u0930\u092e\u0947\u092f\u094b\u0902 \u0915\u093e \u0938\u092e\u093e\u0927\u093e\u0928 \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948, \u091c\u093f\u0938\u092e\u0947\u0902 \u092a\u094d\u0930\u092e\u0947\u092f\u094b\u0902 \u0915\u0940 \u092a\u094d\u0930\u092e\u093e\u0923\u093f\u0915\u0930\u0923 \u0914\u0930 \u0909\u092a\u092a\u0924\u094d\u0924\u093f\u092f\u093e\u0902 \u0936\u093e\u092e\u093f\u0932 \u0939\u0948\u0902\u0964 \u0907\u0928 \u0905\u092d\u094d\u092f\u093e\u0938\u094b\u0902 \u092e\u0947\u0902 \u092a\u0942\u0930\u0915 \u092a\u094d\u0930\u093e\u092f\u093f\u0915\u0924\u093e, \u0938\u0947\u091f\u094b\u0902 \u0915\u093e \u0938\u092e\u094d\u092e\u093f\u0932\u0928 \u0914\u0930 \u0918\u091f\u0928\u093e\u0913\u0902 \u0915\u093e \u0938\u0902\u0915\u0941\u091a\u0928 \u091c\u0948\u0938\u0947 \u0935\u093f\u0937\u092f \u0936\u093e\u092e\u093f\u0932 \u0939\u0948\u0902\u0964 \u0907\u0928 \u0905\u092d\u094d\u092f\u093e\u0938\u094b\u0902 \u0915\u094b \u092a\u0942\u0930\u093e \u0915\u0930\u0928\u0947 \u0938\u0947 \u092a\u094d\u0930\u093e\u092f\u093f\u0915\u0924\u093e \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 \u0915\u093e \u0917\u0939\u0928 \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u090f\u0915 \u092e\u091c\u092c\u0942\u0924 \u0906\u0927\u093e\u0930 \u092e\u093f\u0932\u0947\u0917\u093e\u0964<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>\u0938\u0940\u0916\u0928\u0947 \u0915\u0947 \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f:<\/strong> \u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u0915\u094b \u092a\u0942\u0930\u093e \u0915\u0930\u0928\u0947 \u0915\u0947 \u092c\u093e\u0926 \u091b\u093e\u0924\u094d\u0930 \u0938\u0915\u094d\u0937\u092e \u0939\u094b\u0902\u0917\u0947:\n<\/p>\n<ol>\n<li><strong>\u092a\u094d\u0930\u092e\u093e\u0923\u093f\u0924 \u0915\u0930\u0928\u093e<\/strong> \u092a\u094d\u0930\u093e\u092f\u093f\u0915\u0924\u093e \u0915\u0940 \u092e\u0942\u0932\u092d\u0942\u0924 \u0917\u0941\u0923\u0927\u0930\u094d\u092e<\/li>\n<\/ol>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/SSkPFP5FpKM\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p style=\"text-align: justify; color: #000000;\">\u0906\u0917\u0947 \u0939\u092e \u0915\u0941\u091b \u0905\u092d\u094d\u092f\u093e\u0938 (\u0938\u092e\u093e\u0927\u093e\u0928 \u0915\u093f\u090f \u0917\u090f) \u0926\u0947\u0916\u0947\u0902\u0917\u0947 \u091c\u093f\u0928\u0915\u093e \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f \u092a\u094d\u0930\u093e\u092f\u093f\u0915\u0924\u093e \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 \u0915\u0947 \u0932\u093f\u090f \u0915\u0941\u091b \u0909\u092a\u092f\u094b\u0917\u0940 \u092a\u094d\u0930\u092e\u0947\u092f\u094b\u0902 \u0915\u094b \u092a\u094d\u0930\u092e\u093e\u0923\u093f\u0924 \u0915\u0930\u0928\u093e \u0939\u0948\u0964 \u0915\u094b\u0936\u093f\u0936 \u0915\u0930\u0947\u0902 \u0915\u093f \u0906\u092a \u0909\u0928\u094d\u0939\u0947\u0902 \u0938\u094d\u0935\u092f\u0902 \u0939\u0932 \u0915\u0930\u0947\u0902 \u0914\u0930 \u092b\u093f\u0930 \u0905\u092a\u0928\u0947 \u092a\u0930\u093f\u0923\u093e\u092e\u094b\u0902 \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u0915\u0930\u0947\u0902\u0964<\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><strong><a href=\"https:\/\/www.youtube.com\/watch?v=SSkPFP5FpKM&amp;t=111s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A^c) = 1 -P(A)<\/span> \u0915\u094b \u092a\u094d\u0930\u092e\u093e\u0923\u093f\u0924 \u0915\u0930\u0947\u0902 \u0914\u0930<\/span><\/a> \u0907\u0938\u0938\u0947 \u092f\u0939 \u0924\u0930\u094d\u0915 \u0915\u0930\u0928\u0947 \u0915\u0940 \u090f\u0915 \u0909\u092a\u092a\u0924\u094d\u0924\u093f \u0915\u0930\u0947\u0902 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">P(\\emptyset) = 0<\/span><\/strong><br \/>\n<span class=\"collapseomatic \" id=\"id6a589e6526a8d\"  tabindex=\"0\" title=\"\u0938\u092e\u093e\u0927\u093e\u0928 \u0926\u093f\u0916\u093e\u090f\u0902\"    >\u0938\u092e\u093e\u0927\u093e\u0928 \u0926\u093f\u0916\u093e\u090f\u0902<\/span><div id=\"target-id6a589e6526a8d\" class=\"collapseomatic_content \">\n<p style=\"text-align: justify; color: #000000;\">\u092a\u094d\u0930\u093e\u092f\u093f\u0915\u0924\u093e \u092e\u093e\u092a \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0938\u0947, \u092f\u0926\u093f <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> \u0915\u094b\u0908 \u092d\u0940 \u092e\u093e\u092a \u092f\u094b\u0917\u094d\u092f \u0918\u091f\u0928\u093e\u090f\u0902 \u0939\u0948\u0902, \u0924\u094b \u092f\u0939 \u0938\u0924\u094d\u092f \u0939\u094b\u0917\u093e \u0915\u093f:<\/p>\n<table>\n<tbody>\n<tr>\n<td>[a]<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">0\\leq P(A) \\leq 1<\/span><\/td>\n<\/tr>\n<tr>\n<td>[b]<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">A\\cap B = \\emptyset \\rightarrow P(A\\cup B) = P(A) + P(B)<\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(\\Omega) = 1<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">\u0905\u092c, \u0915\u094d\u092f\u094b\u0902\u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">A\\cap A^c = \\emptyset<\/span>, \u092d\u093e\u0917 [b] \u0938\u0947 \u0939\u092e\u0947\u0902 \u092e\u093f\u0932\u0947\u0917\u093e \u0915\u093f:<\/p>\n<table>\n<tbody>\n<tr>\n<td>[d]<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">A\\cap A^c = \\emptyset \\rightarrow P(A\\cup A^c) = P(A) + P(A^c)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">\u0915\u094d\u092f\u094b\u0902\u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">A\\cap A^c = \\emptyset<\/span> \u0939\u092e\u0947\u0936\u093e \u0938\u0924\u094d\u092f \u0939\u0948 \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">A\\cup A^c = \\Omega<\/span>, \u0924\u094b \u0939\u092e\u0947\u0902 \u092e\u093f\u0932\u0947\u0917\u093e \u0915\u093f<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">1=P(\\Omega) = P(A\\cup A^c) = P(A) + P(A^c)<\/span>\n<p style=\"text-align: justify; color: #000000;\">\u0914\u0930 \u0907\u0938\u0932\u093f\u090f<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A^c) = 1-P(A)<\/span>\n<p style=\"text-align: justify; color: #000000;\">\u091c\u094b \u0939\u092e \u0938\u093e\u092c\u093f\u0924 \u0915\u0930\u0928\u093e \u091a\u093e\u0939\u0924\u0947 \u0925\u0947\u0964<\/p>\n<p style=\"text-align: justify; color: #000000;\">\u092f\u0939 \u0938\u093e\u092c\u093f\u0924 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">P(\\emptyset)=0<\/span>, \u092f\u0939 \u092a\u0930\u094d\u092f\u093e\u092a\u094d\u0924 \u0939\u0948 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">A = \\Omega<\/span> \u0915\u094b \u0909\u0938 \u0938\u0902\u092c\u0902\u0927 \u092e\u0947\u0902 \u0932\u093f\u092f\u093e \u091c\u093e\u090f \u091c\u093f\u0938\u0947 \u0905\u092d\u0940 \u0938\u093e\u092c\u093f\u0924 \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948 \u0914\u0930 \u0939\u092e\u0947\u0902 \u092e\u093f\u0932\u0947\u0917\u093e \u0915\u093f<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(\\emptyset) = P(\\Omega^c) =1 - P(\\Omega) = 1-1 = 0<\/span>\n<\/div><\/li>\n<li><strong><a href=\"https:\/\/www.youtube.com\/watch?v=SSkPFP5FpKM&amp;t=534s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">a) \u092a\u094d\u0930\u092e\u093e\u0923\u093f\u0924 \u0915\u0930\u0947\u0902 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">A\\subseteq B \\rightarrow P(B\\setminus A) = P(B) - P(A)<\/span><\/span><\/a>, \u0915\u094d\u092f\u093e \u092f\u0939 \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0930\u0942\u092a \u0938\u0947 \u0938\u0924\u094d\u092f \u0939\u0948?<\/strong><strong>b) \u092a\u094d\u0930\u092e\u093e\u0923\u093f\u0924 \u0915\u0930\u0947\u0902 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">A\\subseteq B \\rightarrow P(B)\\leq P(A)<\/span><\/strong><span class=\"collapseomatic \" id=\"id6a589e6526cd4\"  tabindex=\"0\" title=\"\u0938\u092e\u093e\u0927\u093e\u0928 \u0926\u093f\u0916\u093e\u090f\u0902 a)\"    >\u0938\u092e\u093e\u0927\u093e\u0928 \u0926\u093f\u0916\u093e\u090f\u0902 a)<\/span><div id=\"target-id6a589e6526cd4\" class=\"collapseomatic_content \">\n<table>\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">A\\subseteq B<\/span>; \u092a\u094d\u0930\u0924\u094d\u092f\u092f<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\equiv A\\cap B = A<\/span><\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">B\\setminus A = B\\cap A^c<\/span>; \u0938\u0947\u091f \u0925\u094d\u092f\u094b\u0930\u0940 \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">B= (B\\cap A) \\cup (B\\cap A^c)<\/span>; \u0938\u0947\u091f\u094b\u0902 \u0915\u0940 \u0935\u093f\u0936\u0947\u0937\u0924\u093e<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(B\\cap A)\\cap (B\\cap A^c)=\\emptyset<\/span>; \u0938\u0947\u091f\u094b\u0902 \u0915\u0940 \u0935\u093f\u0936\u0947\u0937\u0924\u093e<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(B)= P[(B\\cap A) \\cup (B\\cap A^c)]<\/span>; (3) \u0938\u0947<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(B)= P (B\\cap A) + P(B\\cap A^c)<\/span>; (4) + \u092a\u094d\u0930\u093e\u092f\u093f\u0915\u0924\u093e \u092e\u093e\u092a \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0938\u0947<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(B)= P (B\\cap A) + P(B\\setminus A)<\/span>; (2) \u0938\u0947<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> P(B\\setminus A) =P(B) - P (B\\cap A) <\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">{P(B\\setminus A) =P(B) - P (A) }<\/span>; (1) \u0938\u0947<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center; color: #000000;\">\u0907\u0938\u0932\u093f\u090f <span class=\"katex-eq\" data-katex-display=\"false\">{\\{A\\subseteq B\\}\\vdash P(B\\setminus A) = P(B) - P(A)}.<\/span>\n<p style=\"text-align: justify; color: #000000;\">\u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0902 \u0915\u093f \u092f\u0939 \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0930\u0942\u092a \u0938\u0947 \u0938\u0924\u094d\u092f \u0928\u0939\u0940\u0902 \u0939\u0948, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0907\u0938\u0947 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f <span class=\"katex-eq\" data-katex-display=\"false\">A\\subseteq B<\/span> \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964 \u0907\u0928 \u0935\u093f\u0915\u093e\u0938\u094b\u0902 \u0938\u0947 \u092f\u0939 \u0926\u0947\u0916\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948 \u0915\u093f, \u092f\u0926\u093f \u0910\u0938\u093e \u0928\u0939\u0940\u0902 \u0939\u0948, \u0924\u094b \u0939\u092e\u0947\u0902 \u092e\u093f\u0932\u0947\u0917\u093e \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">P(B\\setminus A) = P(B) - P(A\\cap B).<\/span>\n<\/div>\n<span class=\"collapseomatic \" id=\"id6a589e6526e6c\"  tabindex=\"0\" title=\"\u0938\u092e\u093e\u0927\u093e\u0928 \u0926\u093f\u0916\u093e\u090f\u0902 b)\"    >\u0938\u092e\u093e\u0927\u093e\u0928 \u0926\u093f\u0916\u093e\u090f\u0902 b)<\/span><div id=\"target-id6a589e6526e6c\" class=\"collapseomatic_content \">\n<p style=\"text-align: justify; color: #000000;\">a) \u092e\u0947\u0902 \u0926\u093f\u090f \u0917\u090f \u0924\u0930\u094d\u0915 \u0938\u0947 \u0939\u092e\u0947\u0902 \u092e\u093f\u0932\u0947\u0917\u093e \u0915\u093f:<\/p>\n<table>\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> \\{A\\subseteq B\\}\\vdash P(B\\setminus A) = P(B) - P(A)<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\equiv \\{A\\subseteq B\\}\\vdash P(A) + P(B\\setminus A) = P(B)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">\u0905\u0902\u0924 \u092e\u0947\u0902, \u0915\u094d\u092f\u094b\u0902\u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">P<\/span> \u090f\u0915 \u092a\u094d\u0930\u093e\u092f\u093f\u0915\u0924\u093e \u092e\u093e\u092a \u0939\u0948, \u092f\u0939 \u0938\u0924\u094d\u092f \u0939\u0948 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\forall X (P(X)\\geq 0),<\/span> \u0907\u0938\u0932\u093f\u090f, \u092a\u0930\u093f\u0923\u093e\u092e\u0924\u0903:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">{\\{A\\subseteq B\\}\\vdash P(A) \\leq P(B)}.<\/span>\n<\/div>\n<p>&nbsp;<\/li>\n<li><strong><a href=\"https:\/\/www.youtube.com\/watch?v=SSkPFP5FpKM&amp;t=892s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">a) \u092a\u094d\u0930\u092e\u093e\u0923\u093f\u0924 \u0915\u0930\u0947\u0902 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">P(A\\cup B) = P(A) + P(B) - P(A\\cup B).<\/span><\/span><\/a> \u092a\u094d\u0930\u092e\u093e\u0923 \u0915\u0947 \u0938\u093e\u0925 \u090f\u0915 \u0906\u0930\u0947\u0916 \u0936\u093e\u092e\u093f\u0932 \u0915\u0930\u0947\u0902\u0964<\/strong><strong>b) \u092a\u0939\u0932\u0947 \u0915\u0947 \u092a\u0930\u093f\u0923\u093e\u092e \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u092a\u094d\u0930\u092e\u093e\u0923\u093f\u0924 \u0915\u0930\u0947\u0902 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">P(A\\cup B) \\leq P(A) + P(B)<\/span><\/strong><span class=\"collapseomatic \" id=\"id6a589e6526f43\"  tabindex=\"0\" title=\"\u0938\u092e\u093e\u0927\u093e\u0928 \u0926\u093f\u0916\u093e\u090f\u0902 \u092d\u093e\u0917 a)\"    >\u0938\u092e\u093e\u0927\u093e\u0928 \u0926\u093f\u0916\u093e\u090f\u0902 \u092d\u093e\u0917 a)<\/span><div id=\"target-id6a589e6526f43\" class=\"collapseomatic_content \">\n<table>\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">A\\cup B = (A\\triangle B) \\cup (A\\cap B)<\/span>; \u0938\u0947\u091f\u094b\u0902 \u0915\u0940 \u0935\u093f\u0936\u0947\u0937\u0924\u093e<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">A\\triangle B := (A\\setminus B) \\cup (B\\setminus A)<\/span>; \u0938\u092e\u092e\u093f\u0924\u093f \u0905\u0902\u0924\u0930 \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(A\\triangle B)\\cap ( A \\cap B) = \\emptyset<\/span>; \u0938\u0947\u091f\u094b\u0902 \u0915\u0940 \u0935\u093f\u0936\u0947\u0937\u0924\u093e<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(A\\setminus B)\\cap (B\\setminus A) = \\emptyset<\/span>; \u0938\u0947\u091f\u094b\u0902 \u0915\u0940 \u0935\u093f\u0936\u0947\u0937\u0924\u093e<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(A\\cup B) = P[(A\\triangle B) \\cup (A\\cap B)]<\/span>; (1) \u0938\u0947<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(A\\cup B) = P(A\\triangle B) + P(A\\cap B)]<\/span>; (3) \u0938\u0947<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(A\\cup B) = P(A\\setminus B) + P(B\\setminus A) + P(A\\cap B)]<\/span>; (2,4) \u0938\u0947<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(B\\setminus A) = P(B) - P(A\\cap B)<\/span>; \u0905\u092d\u094d\u092f\u093e\u0938 2 \u0915\u093e \u092a\u0930\u093f\u0923\u093e\u092e \u0932\u093e\u0917\u0942 \u0915\u0930\u0928\u093e<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(7)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(A\\setminus B) = P(A) - P(A\\cap B)<\/span>; (6) \u091c\u0948\u0938\u093e \u0939\u0940<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(8)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(A\\cup B) = P(A) - P(A\\cap B) + P(B) - P(A\\cap B) + P(A\\cap B)]<\/span>; (5,6,7) \u0938\u0947<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">{P(A\\cup B) = P(A) + P(B) - P(A\\cap B)}. <\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<span class=\"collapseomatic \" id=\"id6a589e6527091\"  tabindex=\"0\" title=\"\u0938\u092e\u093e\u0927\u093e\u0928 \u0926\u093f\u0916\u093e\u090f\u0902 \u092d\u093e\u0917 b)\"    >\u0938\u092e\u093e\u0927\u093e\u0928 \u0926\u093f\u0916\u093e\u090f\u0902 \u092d\u093e\u0917 b)<\/span><div id=\"target-id6a589e6527091\" class=\"collapseomatic_content \">\n<table>\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash P(A\\cup B) = P(A) + P(B) - P(A\\cap B)<\/span>; \u092d\u093e\u0917 a) \u0915\u093e \u092a\u0930\u093f\u0923\u093e\u092e<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\equiv\\; \\vdash P(A\\cup B) + P(A\\cap B) = P(A) + P(B) <\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\equiv\\; \\vdash {P(A\\cup B) \\leq P(A) + P(B)} <\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/li>\n<li><strong><a href=\"https:\/\/www.youtube.com\/watch?v=SSkPFP5FpKM&amp;t=1625s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u092f\u0926\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\{E_n\\}<\/span> \u0918\u091f\u0928\u093e\u0913\u0902 \u0915\u093e \u090f\u0915 \u0905\u0928\u0902\u0924 \u092a\u0930\u093f\u0935\u093e\u0930 \u0939\u0948<\/span> <\/a>\u0924\u092c <span class=\"katex-eq\" data-katex-display=\"false\">E_1\\supseteq E_2 \\supseteq E_2 \\supseteq \\cdots \\supseteq E_n \\supseteq E_{n+1}\\supseteq \\cdots.<\/span> \u092f\u0939 \u092a\u094d\u0930\u092e\u093e\u0923\u093f\u0924 \u0915\u0930\u0947\u0902 \u0915\u093f<\/strong>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P\\left( \\bigcap_{n=1}^\\infty E_n \\right) = \\lim_{n\\to \\infty}P(E_n)<\/span>\n<span class=\"collapseomatic \" id=\"id6a589e6527154\"  tabindex=\"0\" title=\"\u0938\u092e\u093e\u0927\u093e\u0928 \u0926\u093f\u0916\u093e\u090f\u0902\"    >\u0938\u092e\u093e\u0927\u093e\u0928 \u0926\u093f\u0916\u093e\u090f\u0902<\/span><div id=\"target-id6a589e6527154\" class=\"collapseomatic_content \">\n<table>\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">E_n \\supseteq E_{n+1}<\/span>; \u092a\u0930\u093f\u0915\u0932\u094d\u092a\u0928\u093e<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">E_n^c \\subseteq E_{n+1}^c<\/span>; (1) \u0938\u0947 \u092a\u0942\u0930\u0915\u0924\u093e \u0926\u094d\u0935\u093e\u0930\u093e<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle(E_n^c \\subseteq E_{n+1}^c) \\rightarrow P\\left( \\bigcup_{n=1}^\\infty E_n^c \\right)= \\lim_{n\\to\\infty}P(E_n^c)<\/span>; \u0928\u093f\u0930\u0902\u0924\u0930\u0924\u093e \u0915\u093e \u0917\u0941\u0923<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcup_{n=1}^\\infty E_n^c = \\left( \\bigcap_{n=1}^\\infty E_n \\right)^c<\/span>; \u0921\u0940 \u092e\u0949\u0930\u094d\u0917\u0928 \u0915\u0947 \u0928\u093f\u092f\u092e<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P\\left(E_n^c\\right) = 1 - P(E_n)<\/span>; \u0905\u092d\u094d\u092f\u093e\u0938 1 \u092e\u0947\u0902 \u092a\u094d\u0930\u092e\u093e\u0923\u093f\u0924<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P\\left(\\left[ \\bigcap_{n=1}^\\infty E_n\\right]^c \\right) = \\lim_{n\\to\\infty}[1-P(E_n)] = 1 - \\lim_{n\\to\\infty}P(E_n)<\/span>; (2,4,5) \u0915\u094b (3) \u092a\u0930 \u0932\u093e\u0917\u0942 \u0915\u0930\u0928\u093e<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle 1 - P\\left( \\bigcap_{n=1}^\\infty E_n \\right) = 1 - \\lim_{n\\to\\infty}P(E_n) <\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">{\\displaystyle P\\left( \\bigcap_{n=1}^\\infty E_n \\right) = \\lim_{n\\to\\infty}P(E_n)} <\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\therefore\\{E_n \\supseteq E_{n+1}\\}\\vdash P\\left( \\bigcap_{n=1}^\\infty E_n \\right) = \\lim_{n\\to\\infty}P(E_n).<\/span>\n<p>&nbsp;<\/li>\n<\/ol>\n<\/div>\n<p style=\"text-align: justify; color: #000000;\">\u0907\u0928 \u0905\u092d\u094d\u092f\u093e\u0938\u094b\u0902 \u0915\u094b \u0939\u0932 \u0915\u0930\u0928\u0947 \u0938\u0947 \u0906\u092a \u092a\u094d\u0930\u093e\u092f\u093f\u0915\u0924\u093e \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 \u0915\u093e \u0905\u0927\u094d\u092f\u092f\u0928 \u091c\u093e\u0930\u0940 \u0930\u0916\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u090f\u0915 \u092a\u094d\u0930\u093e\u0930\u0902\u092d\u093f\u0915 \u0906\u0927\u093e\u0930 \u092a\u0942\u0930\u093e \u0915\u0930\u0947\u0902\u0917\u0947\u0964<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u092a\u094d\u0930\u093e\u092f\u093f\u0915\u0924\u093e \u0917\u0923\u0928\u093e \u0915\u0947 \u0932\u093f\u090f \u0909\u092a\u092f\u094b\u0917\u0940 \u092a\u094d\u0930\u092e\u0947\u092f \u0938\u093e\u0930\u093e\u0902\u0936\u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u092e\u0947\u0902, \u0915\u0941\u091b \u092a\u094d\u0930\u093e\u092f\u093f\u0915\u0924\u093e \u0917\u0923\u0928\u093e \u0915\u0947 \u0932\u093f\u090f \u0909\u092a\u092f\u094b\u0917\u0940 \u092a\u094d\u0930\u092e\u0947\u092f\u094b\u0902 \u0915\u093e \u0938\u092e\u093e\u0927\u093e\u0928 \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948, \u091c\u093f\u0938\u092e\u0947\u0902 \u092a\u094d\u0930\u092e\u0947\u092f\u094b\u0902 \u0915\u0940 \u092a\u094d\u0930\u092e\u093e\u0923\u093f\u0915\u0930\u0923 \u0914\u0930 \u0909\u092a\u092a\u0924\u094d\u0924\u093f\u092f\u093e\u0902 \u0936\u093e\u092e\u093f\u0932 \u0939\u0948\u0902\u0964 \u0907\u0928 \u0905\u092d\u094d\u092f\u093e\u0938\u094b\u0902 \u092e\u0947\u0902 \u092a\u0942\u0930\u0915 \u092a\u094d\u0930\u093e\u092f\u093f\u0915\u0924\u093e, \u0938\u0947\u091f\u094b\u0902 \u0915\u093e \u0938\u092e\u094d\u092e\u093f\u0932\u0928 \u0914\u0930 \u0918\u091f\u0928\u093e\u0913\u0902 \u0915\u093e \u0938\u0902\u0915\u0941\u091a\u0928 \u091c\u0948\u0938\u0947 \u0935\u093f\u0937\u092f \u0936\u093e\u092e\u093f\u0932 \u0939\u0948\u0902\u0964 \u0907\u0928 \u0905\u092d\u094d\u092f\u093e\u0938\u094b\u0902 \u0915\u094b \u092a\u0942\u0930\u093e \u0915\u0930\u0928\u0947 \u0938\u0947 \u092a\u094d\u0930\u093e\u092f\u093f\u0915\u0924\u093e \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 \u0915\u093e \u0917\u0939\u0928 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":26394,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":6,"footnotes":""},"categories":[577,678],"tags":[],"class_list":["post-26669","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-577","category-678"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>\u092a\u094d\u0930\u093e\u092f\u093f\u0915\u0924\u093e \u0917\u0923\u0928\u093e \u0915\u0947 \u0932\u093f\u090f \u0909\u092a\u092f\u094b\u0917\u0940 \u092a\u094d\u0930\u092e\u0947\u092f - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"\u0939\u0932 \u0915\u093f\u090f \u0917\u090f \u0905\u092d\u094d\u092f\u093e\u0938\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925 \u092a\u094d\u0930\u0938\u094d\u0924\u0941\u0924 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