Let A = {2, 3, 6, 8, 9, 11} and B = {1, 4, 5, | vedclass

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(C) For the relation $R$ on $A 	imes B$,$(a, b) R (c, d)$ holds if $3ad - 7bc$ is even.$1$. Reflexive: Check if $(a, b) R (a, b)$ holds.$3ab - 7ba =

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reflexive and symmetric but not transitive.

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(C) For the relation $R$ on $A 	imes B$,$(a, b) R (c, d)$ holds if $3ad - 7bc$ is even.$1$. Reflexive: Check if $(a, b) R (a, b)$ holds.$3ab - 7ba = 3ab - 7ab = -4ab$. Since $-4ab$ is always even for any $a \in A, b \in B$,the relation is reflexive.$2$. Symmetric: If $(a, b) R (c, d)$ holds,then $3ad - 7bc$ is even.We need to check if $(c, d) R (a, b)$ holds,i.e.,if $3cb - 7da$ is even.Note that $3cb - 7da = -(3ad - 7bc)$. If $3ad - 7bc$ is even,then $-(3ad - 7bc)$ is also even. Thus,the relation is symmetric.$3$. Transitive: Check if $(a, b) R (c, d)$ and $(c, d) R (e, f)$ implies $(a, b) R (e, f)$.Let $(a, b) = (3, 4)$,$(c, d) = (6, 4)$,and $(e, f) = (3, 1)$.For $(3, 4) R (6, 4)$: $3(3)(4) - 7(4)(6) = 36 - 168 = -132$ (even).For $(6, 4) R (3, 1)$: $3(6)(1) - 7(4)(3) = 18 - 84 = -66$ (even).For $(3, 4) R (3, 1)$: $3(3)(1) - 7(4)(3) = 9 - 84 = -75$ (odd).Since $(3, 4) R (6, 4)$ and $(6, 4) R (3, 1)$ are true,but $(3, 4) R (3, 1)$ is false,the relation is not transitive.Therefore,the relation is reflexive and symmetric but not transitive.

Let $A = \{2, 3, 6, 8, 9, 11\}$ and $B = \{1, 4, 5, 10, 15\}$. Let $R$ be a relation on $A \times B$ defined by $(a, b) R (c, d)$ if and only if $3ad - 7bc$ is an even integer. Then the relation $R$ is

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