Let A = {1, 2, 3, 4, .... ,10} and B = {0, 1, 2, 3, 4}. The number of elements in the relation R = { ( a , b ) A A : 2 ( a - b ) 2 + 3 ( a - b ) B } is __________ . [2023]

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Solution

Q.
Let A = {1, 2, 3, 4, .... ,10} and B = {0, 1, 2, 3, 4}. The number of elements in the relation $R = {(a, b) \in A \times A : 2 {(a - b)}^{2} + 3 (a - b) \in B}$ is __________ . [2023]

Ans.
(18)

$A = {1, 2, 3, 4, \dots, 10}$ ; $B = {0, 1, 2, 3, 4}$

$R =$ ${(a, b) \in A \times A : 2 {(a - b)}^{2} + 3 (a - b) \in B}$

$Now, 2 {(a - b)}^{2} + 3 (a - b) = (a - b) [2 (a - b) + 3] \in B$

$\Rightarrow a = b or a - b = - 2$

$When a = b \Rightarrow 10 ordered pairs$

$When a - b = - 2 \Rightarrow 8 ordered pairs. Total = 18$

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