Let A = {2, 3, 6, 7} and B = {4, 5, 6, 8}. Let R be a relation defined on A × B by ( a 1 , b 1 ) R ( a 2 , b 2 ) if and only if a 1 + a 2 = b 1 + b 2 . Then the number of elements in R is ______________. [2024]

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Solution

Q.
Let A = {2, 3, 6, 7} and B = {4, 5, 6, 8}. Let R be a relation defined on $A \times B$ by $(a_{1}, b_{1}) R (a_{2}, b_{2})$ if and only if $a_{1} + a_{2} = b_{1} + b_{2}$ . Then the number of elements in R is ______________. [2024]

Ans.
(25)

$Given, A = {2, 3, 6, 7} and B = {4, 5, 6, 8}$

$(a_{1}, b_{1}) R (a_{2}, b_{2}) \Rightarrow a_{1} + a_{2} = b_{1} + b_{2}$

$\begin{matrix} (2, 4) R (6, 4) & (3, 6) R (7, 4) \\ (2, 4) R (7, 5) & (3, 5) R (7, 5) \\ (2, 5) R (7, 4) & (6, 5) R (7, 8) \\ (3, 4) R (6, 5) & (6, 8) R (7, 5) \\ (3, 5) R (6, 4) & (7, 6) R (7, 8) \\ (3, 4) R (7, 6) & (6, 4) R (6, 8) \\ (6, 6) R (6, 6) \end{matrix}] \times 2$

Hence, total number of elements = $13 \times 2 - 1 = 25$

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