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 __________.

<p>To find the number of elements in the relation $R$ defined on $A \times B$, we need to determine all pairs $((a_1, b_1), (a_2, b_2))$ such that $a_1 + a_2 = b_1 + b_2$, where $a_1, a_2 \in A$ and $b_1, b_2 \in B$.</p> <p>First, consider all possible sums of pairs from set $A$ and set $B$.</p> <p>Possible sums from set $A = \{2, 3, 6, 7\}$:</p> <ul> <li>$2 + 2 = 4$</li> <li>$2 + 3 = 5$</li> <li>$2 + 6 = 8$</li> <li>$2 + 7 = 9$</li> <li>$3 + 2 = 5$</li> <li>$3 + 3 = 6$</li> <li>$3 + 6 = 9$</li> <li>$3 + 7 = 10$</li> <li>$6 + 2 = 8$</li> <li>$6 + 3 = 9$</li> <li>$6 + 6 = 12$</li> <li>$6 + 7 = 13$</li> <li>$7 + 2 = 9$</li> <li>$7 + 3 = 10$</li> <li>$7 + 6 = 13$</li> <li>$7 + 7 = 14$</li> </ul> <p>Possible sums from set $B = \{4, 5, 6, 8\}$:</p> <ul> <li>$4 + 4 = 8$</li> <li>$4 + 5 = 9$</li> <li>$4 + 6 = 10$</li> <li>$4 + 8 = 12$</li> <li>$5 + 4 = 9$</li> <li>$5 + 5 = 10$</li> <li>$5 + 6 = 11$</li> <li>$5 + 8 = 13$</li> <li>$6 + 4 = 10$</li> <li>$6 + 5 = 11$</li> <li>$6 + 6 = 12$</li> <li>$6 + 8 = 14$</li> <li>$8 + 4 = 12$</li> <li>$8 + 5 = 13$</li> <li>$8 + 6 = 14$</li> <li>$8 + 8 = 16$</li> </ul> <p>Now, identify the common sums from both sets:</p> <p>Common sums: $8, 9, 10, 12, 13, 14$</p> <p>For each common sum, count the pairs from set $A$ and set $B$ that produce these sums:</p> <ul> <li>Sum = 8: From $A$: {(2,6), (6,2)} - 2 pairs; From $B$: {(4,4)} - 1 pair; Hence, 2 * 1 = 2 pairs</li> <li>Sum = 9: From $A$: {(2,7), (3,6), (6,3), (7,2)} - 4 pairs; From $B$: {(4,5), (5,4)} - 2 pairs; Hence, 4 * 2 = 8 pairs</li> <li>Sum = 10: From $A$: {(3,7), (7,3)} - 2 pairs; From $B$: {(4,6), (5,5), (6,4)} - 3 pairs; Hence, 2 * 3 = 6 pairs</li> <li>Sum = 12: From $A$: {(6,6)} - 1 pair; From $B$: {(4,8), (6,6), (8,4)} - 3 pairs; Hence, 1 * 3 = 3 pairs</li> <li>Sum = 13: From $A$: {(6,7), (7,6)} - 2 pairs; From $B$: {(5,8), (8,5)} - 2 pairs; Hence, 2 * 2 = 4 pairs</li> <li>Sum = 14: From $A$: {(7,7)} - 1 pair; From $B$: {(6,8), (8,6)} - 2 pairs; Hence, 1 * 2 = 2 pairs</li> </ul> <p>Adding all these, we get the number of elements in the relation $R$:</p> <p>2 + 8 + 6 + 3 + 4 + 2 = 25</p> <p>Thus, the number of elements in $R$ is 25.</p>