{"id":3073,"date":"2025-06-06T09:46:01","date_gmt":"2025-06-06T09:46:01","guid":{"rendered":"https:\/\/diznr.com\/?p=3073"},"modified":"2025-06-06T09:46:01","modified_gmt":"2025-06-06T09:46:01","slug":"part-11-discrete-mathematics-for-computer-science-in-hindi-equivalence-class-partitions-and","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/part-11-discrete-mathematics-for-computer-science-in-hindi-equivalence-class-partitions-and\/","title":{"rendered":"Part 11-Discrete mathematics for computer science in Hindi- Equivalence class and partitions."},"content":{"rendered":"

Part 11-Discrete mathematics for computer science in Hindi- Equivalence class and partitions.<\/p>\n

[fvplayer id=”239″]<\/p>\n

\u0905\u0932\u0917\u093e\u0924\u094d\u092e\u0915 \u0917\u0923\u093f\u0924 (Discrete Mathematics) \u2013 \u092d\u093e\u0917 11<\/strong><\/h3>\n

\u0938\u092e\u093e\u0928\u0924\u093e \u0935\u0930\u094d\u0917 (Equivalence Class) \u0914\u0930 \u0935\u093f\u092d\u093e\u091c\u0928 (Partitions)<\/strong><\/h3>\n

Discrete Mathematics<\/strong> \u092e\u0947\u0902 \u0938\u092e\u093e\u0928\u0924\u093e \u0935\u0930\u094d\u0917 (Equivalence Class)<\/strong> \u0914\u0930 \u0935\u093f\u092d\u093e\u091c\u0928 (Partition)<\/strong> \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0905\u0935\u0927\u093e\u0930\u0923\u093e\u090f\u0901 \u0939\u0948\u0902, \u091c\u094b \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 (Set Theory) \u0914\u0930 \u0938\u0902\u092c\u0902\u0927\u094b\u0902 (Relations)<\/strong> \u0938\u0947 \u0938\u0902\u092c\u0902\u0927\u093f\u0924 \u0939\u0948\u0902\u0964<\/p>\n

1. \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927 (Equivalence Relation) \u0915\u094d\u092f\u093e \u0939\u0948?<\/strong><\/h3>\n

\u092f\u0926\u093f R<\/strong> \u0915\u094b\u0908 \u0938\u0902\u092c\u0902\u0927 (Relation) \u0939\u094b \u0938\u092e\u0941\u091a\u094d\u091a\u092f (Set) S<\/strong> \u092a\u0930, \u0924\u094b R \u0915\u094b \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927 (Equivalence Relation)<\/strong> \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u092f\u0926\u093f \u0935\u0939 \u0907\u0928 \u0924\u0940\u0928 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u092a\u0942\u0930\u093e \u0915\u0930\u0924\u093e \u0939\u0948:<\/p>\n

(i) \u092a\u094d\u0930\u0924\u094d\u092f\u093e\u0935\u0930\u094d\u0924\u0940\u0924\u093e (Reflexive)<\/strong>:<\/p>\n

\u2200a\u2208S,(a,a)\u2208R\\forall a \\in S, (a, a) \\in R<\/span>\u2200<\/span>a<\/span>\u2208<\/span><\/span>S<\/span>,<\/span>(<\/span>a<\/span>,<\/span>a<\/span>)<\/span>\u2208<\/span><\/span>R<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u0905\u0930\u094d\u0925\u093e\u0924\u094d, \u0939\u0930 \u0924\u0924\u094d\u0935 \u0938\u094d\u0935\u092f\u0902 \u0938\u0947 \u0938\u0902\u092c\u0902\u0927\u093f\u0924 \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f<\/strong>\u0964<\/p>\n

(ii) \u0938\u092e\u092e\u093f\u0924\u0924\u093e (Symmetric)<\/strong>:<\/p>\n

\u2200a,b\u2208S,\u00a0\u092f\u0926\u093f\u00a0(a,b)\u2208R\u00a0\u0924\u094b\u00a0(b,a)\u2208R\\forall a, b \\in S, \\text{ \u092f\u0926\u093f } (a, b) \\in R \\text{ \u0924\u094b } (b, a) \\in R<\/span>\u2200<\/span>a<\/span>,<\/span>b<\/span>\u2208<\/span><\/span>S<\/span>,<\/span>\u00a0<\/span>\u092f\u0926\u093f<\/span>\u00a0<\/span><\/span>(<\/span>a<\/span>,<\/span>b<\/span>)<\/span>\u2208<\/span><\/span>R<\/span>\u00a0<\/span>\u0924\u094b<\/span>\u00a0<\/span><\/span>(<\/span>b<\/span>,<\/span>a<\/span>)<\/span>\u2208<\/span><\/span>R<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u0905\u0930\u094d\u0925\u093e\u0924\u094d, \u092f\u0926\u093f a, b \u0938\u0902\u092c\u0902\u0927\u093f\u0924 \u0939\u0948\u0902, \u0924\u094b b, a \u092d\u0940 \u0938\u0902\u092c\u0902\u0927\u093f\u0924 \u0939\u094b\u0902\u0917\u0947<\/strong>\u0964<\/p>\n

(iii) \u0938\u0902\u0915\u094d\u0930\u093e\u092e\u0915\u0924\u093e (Transitive)<\/strong>:<\/p>\n

\u2200a,b,c\u2208S,\u00a0\u092f\u0926\u093f\u00a0(a,b)\u2208R\u00a0\u0914\u0930\u00a0(b,c)\u2208R\u00a0\u0924\u094b\u00a0(a,c)\u2208R\\forall a, b, c \\in S, \\text{ \u092f\u0926\u093f } (a, b) \\in R \\text{ \u0914\u0930 } (b, c) \\in R \\text{ \u0924\u094b } (a, c) \\in R<\/span>\u2200<\/span>a<\/span>,<\/span>b<\/span>,<\/span>c<\/span>\u2208<\/span><\/span>S<\/span>,<\/span>\u00a0<\/span>\u092f\u0926\u093f<\/span>\u00a0<\/span><\/span>(<\/span>a<\/span>,<\/span>b<\/span>)<\/span>\u2208<\/span><\/span>R<\/span>\u00a0<\/span>\u0914\u0930<\/span>\u00a0<\/span><\/span>(<\/span>b<\/span>,<\/span>c<\/span>)<\/span>\u2208<\/span><\/span>R<\/span>\u00a0<\/span>\u0924\u094b<\/span>\u00a0<\/span><\/span>(<\/span>a<\/span>,<\/span>c<\/span>)<\/span>\u2208<\/span><\/span>R<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u0905\u0930\u094d\u0925\u093e\u0924\u094d, \u092f\u0926\u093f a, b \u0938\u0902\u092c\u0902\u0927\u093f\u0924 \u0939\u0948\u0902 \u0914\u0930 b, c \u092d\u0940 \u0938\u0902\u092c\u0902\u0927\u093f\u0924 \u0939\u0948\u0902, \u0924\u094b a, c \u092d\u0940 \u0938\u0902\u092c\u0902\u0927\u093f\u0924 \u0939\u094b\u0902\u0917\u0947<\/strong>\u0964<\/p>\n

\u092f\u0926\u093f \u0915\u094b\u0908 \u0938\u0902\u092c\u0902\u0927 (Relation) \u0907\u0928 \u0924\u0940\u0928\u094b\u0902 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u092a\u0942\u0930\u093e \u0915\u0930\u0924\u093e \u0939\u0948, \u0924\u094b \u0909\u0938\u0947 \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927 (Equivalence Relation) \u0915\u0939\u0924\u0947 \u0939\u0948\u0902\u0964<\/strong><\/p>\n

2. \u0938\u092e\u093e\u0928\u0924\u093e \u0935\u0930\u094d\u0917 (Equivalence Class) \u0915\u094d\u092f\u093e \u0939\u0948?<\/strong><\/h3>\n

\u092f\u0926\u093f R \u090f\u0915 \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927 \u0939\u0948 S \u092a\u0930<\/strong>, \u0924\u094b \u0939\u0930 \u0924\u0924\u094d\u0935 a \u2208 S<\/strong> \u0915\u093e \u090f\u0915 \u0938\u092e\u093e\u0928\u0924\u093e \u0935\u0930\u094d\u0917 (Equivalence Class)<\/strong> \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948:<\/p>\n

[a]={x\u2208S\u00a0\u2223\u00a0(a,x)\u2208R}[a] = \\{ x \\in S \\ | \\ (a, x) \\in R \\}<\/span>[<\/span>a<\/span>]<\/span>=<\/span><\/span>{<\/span>x<\/span>\u2208<\/span><\/span>S<\/span>\u00a0<\/span>\u2223<\/span>\u00a0<\/span>(<\/span>a<\/span>,<\/span>x<\/span>)<\/span>\u2208<\/span><\/span>R<\/span>}<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u0938\u092e\u093e\u0928\u0924\u093e \u0935\u0930\u094d\u0917 [a] \u0935\u0939 \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u0939\u094b\u0924\u093e \u0939\u0948, \u091c\u093f\u0938\u092e\u0947\u0902 S \u0915\u0947 \u0935\u0947 \u0938\u092d\u0940 \u0924\u0924\u094d\u0935 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902 \u091c\u094b a \u0938\u0947 \u0938\u0902\u092c\u0902\u0927\u093f\u0924 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964<\/strong><\/p>\n

\u0909\u0926\u093e\u0939\u0930\u0923 1: \u0938\u092e\u093e\u0928\u0924\u093e \u0935\u0930\u094d\u0917 \u0915\u0940 \u0917\u0923\u0928\u093e<\/strong><\/h3>\n

\u0938\u092e\u0941\u091a\u094d\u091a\u092f:<\/strong><\/p>\n

S={0,1,2,3,4,5,6}S = \\{0, 1, 2, 3, 4, 5, 6\\}<\/span>S<\/span>=<\/span><\/span>{<\/span>0<\/span>,<\/span>1<\/span>,<\/span>2<\/span>,<\/span>3<\/span>,<\/span>4<\/span>,<\/span>5<\/span>,<\/span>6<\/span>}<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u0938\u0902\u092c\u0902\u0927:<\/strong><\/p>\n

R={(a,b)\u00a0\u2223\u00a0a\u2261b\u00a0(mod\u00a03)}R = \\{ (a, b) \\ | \\ a \\equiv b \\ (\\text{mod } 3) \\}<\/span>R<\/span>=<\/span><\/span>{(<\/span>a<\/span>,<\/span>b<\/span>)<\/span>\u00a0<\/span>\u2223<\/span>\u00a0<\/span>a<\/span>\u2261<\/span><\/span>b<\/span>\u00a0<\/span>(<\/span>mod\u00a0<\/span><\/span>3<\/span>)}<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u00a0\u0907\u0938\u0915\u093e \u092e\u0924\u0932\u092c \u0939\u0948 \u0915\u093f a \u0914\u0930 b \u0915\u093e \u0936\u0947\u0937\u092b\u0932 (remainder) \u0938\u092e\u093e\u0928 \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f \u091c\u092c \u0909\u0928\u094d\u0939\u0947\u0902 3 \u0938\u0947 \u0935\u093f\u092d\u093e\u091c\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u090f\u0964<\/strong><\/p>\n

\u0938\u092e\u093e\u0928\u0924\u093e \u0935\u0930\u094d\u0917 \u0928\u093f\u0915\u093e\u0932\u0947\u0902:<\/strong><\/p>\n

\n\n\n\n\n\n\n\n
\u0935\u0930\u094d\u0917<\/th>\n\u0924\u0924\u094d\u0935<\/th>\n<\/tr>\n<\/thead>\n
[0]<\/td>\n{0, 3, 6}<\/td>\n<\/tr>\n
[1]<\/td>\n{1, 4}<\/td>\n<\/tr>\n
[2]<\/td>\n{2, 5}<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

\u092f\u0939 \u0935\u093f\u092d\u093e\u091c\u0928 (Partition) \u092c\u0928\u093e\u0924\u093e \u0939\u0948, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0939\u0930 \u0924\u0924\u094d\u0935 \u0915\u093f\u0938\u0940 \u0928 \u0915\u093f\u0938\u0940 \u0935\u0930\u094d\u0917 \u092e\u0947\u0902 \u0906 \u091c\u093e\u0924\u093e \u0939\u0948 \u0914\u0930 \u0915\u094b\u0908 \u092d\u0940 \u0926\u094b \u0935\u0930\u094d\u0917 \u0913\u0935\u0930\u0932\u0948\u092a \u0928\u0939\u0940\u0902 \u0939\u094b\u0924\u0947\u0964<\/strong><\/p>\n

3. \u0935\u093f\u092d\u093e\u091c\u0928 (Partition) \u0915\u094d\u092f\u093e \u0939\u0948?<\/strong><\/h3>\n

\u00a0\u0915\u093f\u0938\u0940 \u0938\u092e\u0941\u091a\u094d\u091a\u092f S<\/strong> \u0915\u093e \u0935\u093f\u092d\u093e\u091c\u0928 (Partition)<\/strong>, S \u0915\u0947 \u0917\u0948\u0930-\u0930\u093f\u0915\u094d\u0924 \u0909\u092a\u0938\u092e\u0941\u091a\u094d\u091a\u092f\u094b\u0902 \u0915\u093e \u090f\u0915 \u0938\u092e\u0941\u091a\u094d\u091a\u092f<\/strong> \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u094b \u0907\u0928 \u0936\u0930\u094d\u0924\u094b\u0902 \u0915\u094b \u092a\u0942\u0930\u093e \u0915\u0930\u0924\u093e \u0939\u0948:<\/p>\n

\u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0924\u0924\u094d\u0935 \u0915\u093f\u0938\u0940 \u090f\u0915 \u0909\u092a\u0938\u092e\u0941\u091a\u094d\u091a\u092f \u092e\u0947\u0902 \u091c\u0930\u0942\u0930 \u0939\u094b\u0964<\/strong>
\u0915\u094b\u0908 \u092d\u0940 \u0926\u094b \u0909\u092a\u0938\u092e\u0941\u091a\u094d\u091a\u092f \u0913\u0935\u0930\u0932\u0948\u092a \u0928\u0939\u0940\u0902 \u0915\u0930\u0924\u0947\u0964<\/strong>
\u0938\u092d\u0940 \u0909\u092a\u0938\u092e\u0941\u091a\u094d\u091a\u092f\u094b\u0902 \u0915\u093e \u0938\u0902\u092f\u094b\u091c\u0928 (Union) \u092a\u0942\u0930\u0947 S \u0915\u094b \u092c\u0928\u093e\u0924\u093e \u0939\u0948\u0964<\/strong><\/p>\n

\u0909\u0926\u093e\u0939\u0930\u0923 2: \u0935\u093f\u092d\u093e\u091c\u0928 \u092c\u0928\u093e\u0928\u093e<\/strong><\/h3>\n

\u0938\u092e\u0941\u091a\u094d\u091a\u092f:<\/strong><\/p>\n

S={1,2,3,4,5,6}S = \\{1, 2, 3, 4, 5, 6\\}<\/span>S<\/span>=<\/span><\/span>{<\/span>1<\/span>,<\/span>2<\/span>,<\/span>3<\/span>,<\/span>4<\/span>,<\/span>5<\/span>,<\/span>6<\/span>}<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u0938\u0902\u092d\u093e\u0935\u093f\u0924 \u0935\u093f\u092d\u093e\u091c\u0928:<\/strong><\/p>\n

P1={{1,2},{3,4},{5,6}}P_1 = \\{\\{1, 2\\}, \\{3, 4\\}, \\{5, 6\\}\\}<\/span>P<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>{{<\/span>1<\/span>,<\/span>2<\/span>}<\/span>,<\/span>{<\/span>3<\/span>,<\/span>4<\/span>}<\/span>,<\/span>{<\/span>5<\/span>,<\/span>6<\/span>}}<\/span><\/span><\/span><\/span><\/span> P2={{1,3,5},{2,4,6}}P_2 = \\{\\{1, 3, 5\\}, \\{2, 4, 6\\}\\}<\/span>P<\/span>2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>{{<\/span>1<\/span>,<\/span>3<\/span>,<\/span>5<\/span>}<\/span>,<\/span>{<\/span>2<\/span>,<\/span>4<\/span>,<\/span>6<\/span>}}<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u092f\u0939 \u0926\u094b\u0928\u094b\u0902 \u0935\u093f\u092d\u093e\u091c\u0928 \u0938\u0939\u0940 \u0939\u0948\u0902 \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0924\u0924\u094d\u0935 \u0915\u094b \u0915\u0935\u0930 \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948 \u0914\u0930 \u0915\u094b\u0908 \u092d\u0940 \u0926\u094b \u0909\u092a\u0938\u092e\u0941\u091a\u094d\u091a\u092f \u0913\u0935\u0930\u0932\u0948\u092a \u0928\u0939\u0940\u0902 \u0915\u0930 \u0930\u0939\u0947 \u0939\u0948\u0902\u0964<\/strong><\/p>\n

4. \u0938\u092e\u093e\u0928\u0924\u093e \u0935\u0930\u094d\u0917 \u0914\u0930 \u0935\u093f\u092d\u093e\u091c\u0928 \u092e\u0947\u0902 \u0938\u0902\u092c\u0902\u0927<\/strong><\/h3>\n

\u00a0\u092f\u0926\u093f \u0915\u094b\u0908 \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927 (Equivalence Relation) \u0926\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u094b<\/strong>, \u0924\u094b \u0935\u0939 S \u0915\u094b \u0938\u092e\u093e\u0928\u0924\u093e \u0935\u0930\u094d\u0917\u094b\u0902 \u092e\u0947\u0902 \u0935\u093f\u092d\u093e\u091c\u093f\u0924 \u0915\u0930\u0924\u093e \u0939\u0948<\/strong>\u0964
\u0938\u092e\u093e\u0928\u0924\u093e \u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u093e \u0938\u0902\u0918 (Union) \u092a\u0942\u0930\u093e S \u0926\u0947\u0924\u093e \u0939\u0948<\/strong>, \u0914\u0930 \u0935\u0947 \u0913\u0935\u0930\u0932\u0948\u092a \u0928\u0939\u0940\u0902 \u0915\u0930\u0924\u0947, \u0907\u0938\u0932\u093f\u090f \u0935\u0947 S \u0915\u093e \u090f\u0915 \u0935\u093f\u092d\u093e\u091c\u0928 \u092c\u0928\u093e\u0924\u0947 \u0939\u0948\u0902<\/strong>\u0964<\/p>\n

\u0907\u0938\u0915\u093e \u092e\u0924\u0932\u092c \u0939\u0948 \u0915\u093f \u0939\u0930 \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927, S \u0915\u0947 \u090f\u0915 \u0935\u093f\u092d\u093e\u091c\u0928 \u0915\u094b \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0924\u093e \u0939\u0948\u0964<\/strong><\/p>\n

5. \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u092a\u094d\u0930\u0936\u094d\u0928 (GATE & UGC NET Level)<\/strong><\/h3>\n

Q1: \u092f\u0926\u093f \u0938\u092e\u0941\u091a\u094d\u091a\u092f S = {1, 2, 3, 4, 5, 6} \u092a\u0930 \u0938\u0902\u092c\u0902\u0927 R \u0926\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948:<\/strong><\/h3>\n

R={(a,b)\u2223a\u2212b\u00a0\u090f\u0915\u00a0\u0938\u092e\u00a0\u0938\u0902\u0916\u094d\u092f\u093e\u00a0\u0939\u0948}R = \\{(a, b) | a – b \\text{ \u090f\u0915 \u0938\u092e \u0938\u0902\u0916\u094d\u092f\u093e \u0939\u0948} \\}<\/span>R<\/span>=<\/span><\/span>{(<\/span>a<\/span>,<\/span>b<\/span>)<\/span>\u2223<\/span>a<\/span>\u2212<\/span><\/span>b<\/span>\u00a0<\/span>\u090f\u0915<\/span>\u00a0<\/span>\u0938\u092e<\/span>\u00a0<\/span>\u0938\u0902\u0916\u094d\u092f\u093e<\/span>\u00a0<\/span>\u0939\u0948<\/span><\/span>}<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u0924\u094b \u0907\u0938\u0915\u0947 \u0938\u092e\u093e\u0928\u0924\u093e \u0935\u0930\u094d\u0917 \u0915\u094d\u092f\u093e \u0939\u094b\u0902\u0917\u0947?<\/p>\n

\u0909\u0924\u094d\u0924\u0930:<\/strong>
\u00a0\u092f\u0926\u093f a\u2212ba – b<\/span>a<\/span>\u2212<\/span><\/span>b<\/span><\/span><\/span><\/span> \u0938\u092e (even) \u0939\u0948, \u0924\u094b a \u0914\u0930 b \u0915\u093e \u0905\u0902\u0924\u0930 \u0938\u092e \u0938\u0902\u0916\u094d\u092f\u093e \u0939\u094b\u0917\u093e, \u092f\u093e\u0928\u0940 \u0935\u0947 \u090f\u0915 \u0939\u0940 \u0935\u0930\u094d\u0917 \u092e\u0947\u0902 \u0939\u094b\u0902\u0917\u0947\u0964
\u0938\u092e\u093e\u0928\u0924\u093e \u0935\u0930\u094d\u0917:<\/p>\n

[1]={1,3,5},[2]={2,4,6}[1] = \\{1, 3, 5\\}, \\quad [2] = \\{2, 4, 6\\}<\/span>[<\/span>1<\/span>]<\/span>=<\/span><\/span>{<\/span>1<\/span>,<\/span>3<\/span>,<\/span>5<\/span>}<\/span>,<\/span>[<\/span>2<\/span>]<\/span>=<\/span><\/span>{<\/span>2<\/span>,<\/span>4<\/span>,<\/span>6<\/span>}<\/span><\/span><\/span><\/span><\/span><\/p>\n

6. \u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937 (Conclusion)<\/strong><\/h3>\n

\u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927<\/strong> \u0935\u0939 \u0938\u0902\u092c\u0902\u0927 \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u094b \u092a\u094d\u0930\u0924\u094d\u092f\u093e\u0935\u0930\u094d\u0924\u0940, \u0938\u092e\u092e\u093f\u0924 \u0914\u0930 \u0938\u0902\u0915\u094d\u0930\u093e\u092e\u0915<\/strong> \u0939\u094b\u0924\u093e \u0939\u0948\u0964
\u0938\u092e\u093e\u0928\u0924\u093e \u0935\u0930\u094d\u0917<\/strong> \u090f\u0915 \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u0915\u0947 \u0909\u0928 \u0938\u092d\u0940 \u0924\u0924\u094d\u0935\u094b\u0902 \u0915\u093e \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u0939\u094b\u0924\u093e \u0939\u0948, \u091c\u094b \u090f\u0915 \u0935\u093f\u0936\u0947\u0937 \u0924\u0924\u094d\u0935 \u0938\u0947 \u0938\u0902\u092c\u0902\u0927\u093f\u0924 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964
\u0935\u093f\u092d\u093e\u091c\u0928<\/strong> \u090f\u0915 \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u0915\u0947 \u0917\u0948\u0930-\u0913\u0935\u0930\u0932\u0948\u092a\u093f\u0902\u0917 \u0909\u092a\u0938\u092e\u0941\u091a\u094d\u091a\u092f\u094b\u0902<\/strong> \u0915\u093e \u090f\u0915 \u0938\u092e\u0942\u0939 \u0939\u094b\u0924\u093e \u0939\u0948, \u091c\u093f\u0928\u0915\u093e \u0938\u0902\u0918 \u092a\u0942\u0930\u0947 \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u0915\u094b \u0926\u0947\u0924\u093e \u0939\u0948\u0964
\u0939\u0930 \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927 \u090f\u0915 \u0935\u093f\u092d\u093e\u091c\u0928 \u092c\u0928\u093e\u0924\u093e \u0939\u0948, \u0914\u0930 \u0939\u0930 \u0935\u093f\u092d\u093e\u091c\u0928 \u090f\u0915 \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927 \u0926\u0930\u094d\u0936\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/strong><\/p>\n

\u00a0\u0915\u094d\u092f\u093e \u0906\u092a \u0914\u0930 \u0909\u0926\u093e\u0939\u0930\u0923 \u091a\u093e\u0939\u0924\u0947 \u0939\u0948\u0902? \u092f\u093e GATE\/NET \u0915\u0947 \u0932\u093f\u090f \u0914\u0930 \u092a\u094d\u0930\u0936\u094d\u0928\u094b\u0902 \u0915\u0940 \u0935\u094d\u092f\u093e\u0916\u094d\u092f\u093e \u091a\u093e\u0939\u093f\u090f?\u00a0<\/strong><\/h3>\n

\u0921\u093f\u0938\u094d\u0915\u094d\u0930\u0940\u091f \u0917\u0923\u093f\u0924<\/strong> \u092e\u0947\u0902 \u0938\u092e\u093e\u0928\u0924\u093e \u0935\u0930\u094d\u0917 (Equivalence Class)<\/strong> \u0914\u0930 \u0935\u093f\u092d\u093e\u091c\u0928 (Partition)<\/strong> \u0926\u094b \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0905\u0935\u0927\u093e\u0930\u0923\u093e\u090f\u0901 \u0939\u0948\u0902 \u091c\u094b \u0938\u0947\u091f \u0925\u094d\u092f\u094b\u0930\u0940 \u0914\u0930 \u0938\u0902\u092c\u0902\u0927\u094b\u0902 (Relations) \u0915\u0940 \u0917\u0939\u0930\u093e\u0908 \u0938\u0947 \u0938\u092e\u091d \u092a\u094d\u0930\u0926\u093e\u0928 \u0915\u0930\u0924\u0940 \u0939\u0948\u0902\u0964<\/span><\/p>\n

\n

\ud83d\udd39 \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927 (Equivalence Relation)<\/h2>\n

\u090f\u0915 \u0938\u0902\u092c\u0902\u0927 RR<\/span>R<\/span><\/span><\/span><\/span> \u0915\u094b \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927<\/strong> \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u092f\u0926\u093f \u0935\u0939 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0924\u0940\u0928 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0915\u0930\u0924\u093e \u0939\u0948:<\/span><\/p>\n

    \n
  1. \n

    \u092a\u094d\u0930\u0924\u093f\u092b\u0932\u0924\u093e (Reflexive):<\/strong> \u0939\u0930 \u0924\u0924\u094d\u0935 \u0938\u094d\u0935\u092f\u0902 \u0938\u0947 \u0938\u0902\u092c\u0902\u0927\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/span>
    \u0909\u0926\u093e\u0939\u0930\u0923: a\u223caa \\sim a<\/span>a<\/span>\u223c<\/span><\/span>a<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n

  2. \n

    \u0938\u092e\u092e\u093f\u0924\u093f (Symmetric):<\/strong> \u092f\u0926\u093f a\u223cba \\sim b<\/span>a<\/span>\u223c<\/span><\/span>b<\/span><\/span><\/span><\/span>, \u0924\u094b b\u223cab \\sim a<\/span>b<\/span>\u223c<\/span><\/span>a<\/span><\/span><\/span><\/span> \u092d\u0940 \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964<\/span><\/p>\n<\/li>\n

  3. \n

    \u0938\u093e\u0902\u0915\u094d\u0930\u093e\u092e\u0915\u0924\u093e (Transitive):<\/strong> \u092f\u0926\u093f a\u223cba \\sim b<\/span>a<\/span>\u223c<\/span><\/span>b<\/span><\/span><\/span><\/span> \u0914\u0930 b\u223ccb \\sim c<\/span>b<\/span>\u223c<\/span><\/span>c<\/span><\/span><\/span><\/span>, \u0924\u094b a\u223cca \\sim c<\/span>a<\/span>\u223c<\/span><\/span>c<\/span><\/span><\/span><\/span> \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964<\/span><\/p>\n<\/li>\n<\/ol>\n

    \n

    \ud83d\udd39 \u0938\u092e\u093e\u0928\u0924\u093e \u0935\u0930\u094d\u0917 (Equivalence Class)<\/h2>\n

    \u092f\u0926\u093f RR<\/span>R<\/span><\/span><\/span><\/span> \u090f\u0915 \u0938\u092e\u093e\u0928\u0924\u093e \u0938\u0902\u092c\u0902\u0927 \u0939\u0948 \u0914\u0930 aa<\/span>a<\/span><\/span><\/span><\/span> \u090f\u0915 \u0924\u0924\u094d\u0935 \u0939\u0948, \u0924\u094b aa<\/span>a<\/span><\/span><\/span><\/span> \u0915\u093e \u0938\u092e\u093e\u0928\u0924\u093e \u0935\u0930\u094d\u0917<\/strong> [a][a]<\/span>[<\/span>a<\/span>]<\/span><\/span><\/span><\/span> \u0909\u0928 \u0938\u092d\u0940 \u0924\u0924\u094d\u0935\u094b\u0902 \u0915\u093e \u0938\u0947\u091f \u0939\u0948 \u091c\u094b aa<\/span>a<\/span><\/span><\/span><\/span> \u0915\u0947 \u0938\u092e\u093e\u0928 \u0939\u0948\u0902:<\/span><\/p>\n

    [a]={x\u2208S\u2223x\u223ca}[a] = \\{ x \\in S \\mid x \\sim a \\}<\/span>[<\/span>a<\/span>]<\/span>=<\/span><\/span>{<\/span>x<\/span>\u2208<\/span><\/span>S<\/span>\u2223<\/span><\/span>x<\/span>\u223c<\/span><\/span>a<\/span>}<\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u0909\u0926\u093e\u0939\u0930\u0923:<\/strong> \u092f\u0926\u093f \u0939\u092e \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u093e \u0938\u0947\u091f Z\\mathbb{Z}<\/span>Z<\/span><\/span><\/span><\/span> \u0932\u0947\u0902 \u0914\u0930 \u0938\u0902\u092c\u0902\u0927 a\u223cba \\sim b<\/span>a<\/span>\u223c<\/span><\/span>b<\/span><\/span><\/span><\/span> \u0915\u094b \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0947\u0902 \u092f\u0926\u093f a\u2212ba – b<\/span>a<\/span>\u2212<\/span><\/span>b<\/span><\/span><\/span><\/span> 3 \u0938\u0947 \u0935\u093f\u092d\u093e\u091c\u094d\u092f \u0939\u094b, \u0924\u094b:<\/span><\/p>\n