Question 85

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

Solution

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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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 _________

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