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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 _________
Correct Answer: 25
| Sum (s) | Pairs in A×A with sum s | Count NA(s) | Pairs in B×B with sum s | Count NB(s) | Contribution (NA×NB) |
| 8 | $$(2,6), (6,2)$$ | 2 | $$(4,4)$$ | 1 | $$2 \times 1 = 2$$ |
| 9 | $$(2,7), (7,2), (3,6), (6,3)$$ | 4 | $$(4,5), (5,4)$$ | 2 | $$4 \times 2 = 8$$ |
| 10 | $$(3,7), (7,3)$$ | 2 | $$(4,6), (6,4), (5,5)$$ | 3 | $$2 \times 3 = 6$$ |
| 12 | $$(6,6)$$ | 1 | $$(4,8), (8,4), (6,6)$$ | 3 | $$1 \times 3 = 3$$ |
| 13 | $$(6,7), (7,6)$$ | 2 | $$(5,8), (8,5)$$ | 2 | $$2 \times 2 = 4$$ |
| 14 | $$(7,7)$$ | 1 | $$(6,8), (8,6)$$ | 2 | $$1 \times 2 = 2$$ |
$$\text{Total elements in } R = 2 + 8 + 6 + 3 + 4 + 2 = 25$$
The relation $$R$$ contains exactly 25 elements.
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