Let 𝐴 = {1,3,4,6,9} and 𝐵 = {2,4,5,8,10}. Let 𝑅 be a relation defined on 𝐴 × 𝐵 such that 𝑅 = {((𝑎1, 𝑏1), (𝑎2, 𝑏2))∶ 𝑎1 ≤ 𝑏2 and 𝑏1 ≤ 𝑎2}. Then the number of elements in the set 𝑅 is

Question

Let 𝐴 = 1,3,4,6,9 and 𝐵 = 2,4,5,8,10. Let 𝑅 be a relation defined on 𝐴 × 𝐵 such that 𝑅 = ((𝑎1, 𝑏1), (𝑎2, 𝑏2))∶ 𝑎1 ≤ 𝑏2 and 𝑏1 ≤ 𝑎2. Then the number of elements in the set 𝑅 is

DivineJEE · Accepted Answer

(d): We have, 𝐴 = 1,3,4,6,9; 𝐵 = 2,4,5,8,10 nn𝑅 = ((𝑎1, 𝑏1), (𝑎2, 𝑏2))∶ 𝑎1 ≤ 𝑏2 and 𝑏1 ≤ 𝑎2 nnAt 𝑎1 = 1, there are 5 choices for 𝑏2; nn𝑎1 = 3, there are 4 choices for 𝑏2; nn𝑎1 = 4, there are 4 choices for 𝑏2; nn𝑎1 = 6, there are 2 choices for 𝑏2; nn𝑎1 = 9, there are 1 choice for 𝑏2; nn∴ Total ways for (𝑎1 ≤ 𝑏2) = 16 nnNow, at 𝑏1 = 2, there are 4 choices for 𝑎2; nn𝑏1 = 4, there are 3 choices for 𝑎2; nn𝑏1 = 5, there are 2 choices for 𝑎2; nn𝑏1 = 8, there are 1 choice for 𝑎2 nnTotal ways for (𝑏1 ≤ 𝑎2) = 10 nn∴ Required number ways = 16 × 10 = 160

Explanation

If 𝑅 is the smallest equivalence relation on the set {1,2,3,4} such that {(1,2), (1,3)} ⊂ 𝑅, then the number of elements in 𝑅 is .

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