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(C) The relation $R$ is defined on the set $S = \{1, 2, 3, 4\} 	imes \{1, 2, 3, 4\}$. The total number of elements in $S$ is $4 	imes 4 = 16$. The c

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$12$

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(C) The relation $R$ is defined on the set $S = \{1, 2, 3, 4\} 	imes \{1, 2, 3, 4\}$. The total number of elements in $S$ is $4 	imes 4 = 16$. The condition for the relation is $2a + 3b = 3c + 4d$,where $a, b, c, d \in \{1, 2, 3, 4\}$.We calculate the value of $f(a, b) = 2a + 3b$ for all pairs $(a, b)$:$(1,1) 	o 5, (1,2) 	o 8, (1,3) 	o 11, (1,4) 	o 14$$(2,1) 	o 7, (2,2) 	o 10, (2,3) 	o 13, (2,4) 	o 16$$(3,1) 	o 9, (3,2) 	o 12, (3,3) 	o 15, (3,4) 	o 18$$(4,1) 	o 11, (4,2) 	o 14, (4,3) 	o 17, (4,4) 	o 20$Now calculate $g(c, d) = 3c + 4d$ for all pairs $(c, d)$:$(1,1) 	o 7, (1,2) 	o 11, (1,3) 	o 15, (1,4) 	o 19$$(2,1) 	o 10, (2,2) 	o 14, (2,3) -> 18, (2,4) -> 22$$(3,1) 	o 13, (3,2) 	o 17, (3,3) 	o 21, (3,4) 	o 25$$(4,1) 	o 16, (4,2) 	o 20, (4,3) 	o 24, (4,4) 	o 28$Matching values of $f(a, b) = g(c, d)$:$11: (1,3) 	ext{ and } (4,1) 	ext{ map to } (1,2)$$14: (1,4) 	ext{ and } (4,2) 	ext{ map to } (2,2)$$16: (2,4) 	ext{ maps to } (4,1)$$7: (2,1) 	ext{ maps to } (1,1)$$10: (2,2) 	ext{ maps to } (2,1)$$13: (2,3) 	ext{ maps to } (3,1)$$15: (3,3) 	ext{ maps to } (1,3)$$18: (3,4) 	ext{ maps to } (2,3)$$17: (4,3) 	ext{ maps to } (3,2)$$20: (4,4) 	ext{ maps to } (4,2)$Counting the pairs $((a,b), (c,d))$ satisfying the condition,we find $12$ such elements.

Let $R$ be a relation defined on the set $\{1,2,3,4\} \times \{1,2,3,4\}$ by $R = \{((a,b), (c,d)) : 2a + 3b = 3c + 4d\}$. Then the number of elements in $R$ is:

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