Day 02-Discrete mathematics for gate – Concept of Cartesian product and Introduction of Relation.<\/p>\n
[fvplayer id=”261″]<\/p>\n
Here\u2019s a clear explanation of Day 02: Discrete Mathematics for GATE<\/strong> focusing on the Concept of Cartesian Product<\/strong> and an Introduction to Relations<\/strong>, tailored for GATE CSE\/IT aspirants<\/strong>. \u2705 Bilingual format (English + Hindi)<\/strong> for better understanding.<\/p>\n Cartesian Product (\u0915\u093e\u0930\u094d\u0924\u0940\u092f \u0917\u0941\u0923\u0928\u092b\u0932)<\/p>\n<\/li>\n Introduction to Relations (\u0938\u0902\u092c\u0902\u0927 \u0915\u093e \u092a\u0930\u093f\u091a\u092f)<\/p>\n<\/li>\n<\/ul>\n The Cartesian product<\/strong> of two sets A and B, written as A \u00d7 B<\/strong>, is the set of all ordered pairs<\/strong> (a, b) where a \u2208 A<\/strong> and b \u2208 B<\/strong>.<\/p>\n \u0926\u094b \u0938\u0947\u091f A \u0914\u0930 B \u0915\u093e Cartesian Product<\/strong>, \u092f\u093e\u0928\u093f A \u00d7 B<\/strong>, \u0909\u0928 \u0938\u092d\u0940 ordered pairs (a, b) \u0915\u093e \u0938\u0947\u091f \u0939\u0948 \u091c\u0939\u093e\u0901 a \u0938\u0947\u091f A \u0938\u0947 \u0939\u0948<\/strong> \u0914\u0930 b \u0938\u0947\u091f B \u0938\u0947 \u0939\u0948<\/strong>\u0964<\/p>\n A\u00d7B={(a,b)\u2223a\u2208A,\u00a0b\u2208B}A \u00d7 B = \\{ (a, b) \\mid a \\in A,\\ b \\in B \\}<\/span>A<\/span>\u00d7<\/span><\/span>B<\/span>=<\/span><\/span>{(<\/span>a<\/span>,<\/span>b<\/span>)<\/span>\u2223<\/span><\/span>a<\/span>\u2208<\/span><\/span>A<\/span>,<\/span>\u00a0<\/span>b<\/span>\u2208<\/span><\/span>B<\/span>}<\/span><\/span><\/span><\/span><\/span><\/p>\n Let: Then:<\/p>\n A\u00d7B={(1,x),(1,y),(2,x),(2,y)}A \u00d7 B = \\{(1, x), (1, y), (2, x), (2, y)\\}<\/span>A<\/span>\u00d7<\/span><\/span>B<\/span>=<\/span><\/span>{(<\/span>1<\/span>,<\/span>x<\/span>)<\/span>,<\/span>(<\/span>1<\/span>,<\/span>y<\/span>)<\/span>,<\/span>(<\/span>2<\/span>,<\/span>x<\/span>)<\/span>,<\/span>(<\/span>2<\/span>,<\/span>y<\/span>)}<\/span><\/span><\/span><\/span><\/span> \u2223A\u00d7B\u2223=\u2223A\u2223\u00d7\u2223B\u2223=2\u00d72=4|A \u00d7 B| = |A| \u00d7 |B| = 2 \u00d7 2 = 4<\/span>\u2223<\/span>A<\/span>\u00d7<\/span><\/span>B<\/span>\u2223<\/span>=<\/span><\/span>\u2223<\/span>A<\/span>\u2223<\/span>\u00d7<\/span><\/span>\u2223<\/span>B<\/span>\u2223<\/span>=<\/span><\/span>2<\/span>\u00d7<\/span><\/span>2<\/span>=<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/p>\n A \u00d7 B \u2260 B \u00d7 A (Order matters)<\/p>\n<\/li>\n If A or B is empty, A \u00d7 B is also empty.<\/p>\n<\/li>\n<\/ul>\n A relation<\/strong> R from set A to set B is a subset of the Cartesian product<\/strong> A \u00d7 B.<\/p>\n That is,<\/p>\n R\u2286A\u00d7BR \\subseteq A \u00d7 B<\/span>R<\/span>\u2286<\/span><\/span>A<\/span>\u00d7<\/span><\/span>B<\/span><\/span><\/span><\/span><\/span><\/p>\n Relation (\u0938\u0902\u092c\u0902\u0927) \u090f\u0915 \u0910\u0938\u093e \u0938\u0947\u091f \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u094b A \u00d7 B<\/strong> \u0915\u0947 \u0915\u0941\u091b selected ordered pairs \u0915\u094b \u091a\u0941\u0928\u0924\u093e \u0939\u0948\u0964 \u092f\u0939 \u090f\u0915 subset \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/p>\n Let: R={(1,2),\u00a0(1,3),\u00a0(2,3)}R = \\{(1, 2),\\ (1, 3),\\ (2, 3)\\}<\/span>R<\/span>=<\/span><\/span>{(<\/span>1<\/span>,<\/span>2<\/span>)<\/span>,<\/span>\u00a0<\/span>(<\/span>1<\/span>,<\/span>3<\/span>)<\/span>,<\/span>\u00a0<\/span>(<\/span>2<\/span>,<\/span>3<\/span>)}<\/span><\/span><\/span><\/span><\/span><\/p>\n \u092f\u0939 relation R<\/strong> A \u0938\u0947 B \u092a\u0930 define \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948, \u0914\u0930 \u092f\u0939 A \u00d7 B \u0915\u093e \u090f\u0915 subset \u0939\u0948\u0964<\/p>\n
\n\ud83d\udcd8 Day 02 \u2013 Discrete Mathematics for GATE<\/strong><\/h2>\n
\ud83d\udccc Topics Covered:<\/h3>\n
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\n\ud83d\udd39 1. Cartesian Product \u2013 Concept<\/strong><\/h2>\n
\ud83d\udd38 English Definition:<\/h3>\n
\ud83d\udd38 Hindi Meaning:<\/h3>\n
\n\u2705 Formula:<\/strong><\/h3>\n
\n\ud83e\udde0 Example:<\/strong><\/h3>\n
A = {1, 2}
B = {x, y}<\/p>\n
\n\ud83d\udccc Important Notes:<\/h3>\n
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\n\ud83d\udd39 2. Introduction to Relations<\/strong><\/h2>\n
\ud83d\udd38 English Definition:<\/h3>\n
\ud83d\udd38 Hindi Explanation:<\/h3>\n
\n\u2705 Example of Relation:<\/strong><\/h3>\n
A = {1, 2, 3}
Define relation R = \u201ca is less than b\u201d
Then B = A, and:<\/p>\n
\n\ud83d\udd22 Types of Relations<\/strong> (for upcoming sessions):<\/h2>\n