Let a relation 𝑅 on 𝑁 × 𝑁 be defined as: (𝑥1, 𝑦1)𝑅(𝑥2, 𝑦2) if and only if 𝑥1 ≤ 𝑥2 or 𝑦1 ≤ 𝑦2. Consider the two statements : (I) 𝑅 is reflexive but not symmetric. (II) 𝑅 is transitive Then which one of the following is true?

Question

Let a relation 𝑅 on 𝑁 × 𝑁 be defined as: (𝑥1, 𝑦1)𝑅(𝑥2, 𝑦2) if and only ifn𝑥1 ≤ 𝑥2 or 𝑦1 ≤ 𝑦2. Consider the two statements :n(I) 𝑅 is reflexive but not symmetric.n(II) 𝑅 is transitivenThen which one of the following is true?nn

DivineJEE · Accepted Answer

(𝑥1, 𝑦1)𝑅(𝑥2, 𝑦2)n⇒ 𝑥1 ≤ 𝑥2 or 𝑦1 ≤ 𝑦2nFor reflexive :n(𝑥1, 𝑦1)(𝑥1, 𝑦1)n⇒ 𝑥1 ≤ 𝑥1 or 𝑦1 ≤ 𝑦1 which is true. So, 𝑅 is reflexive.nFor symmetric :nWhen (𝑥1, 𝑦1)(𝑥2, 𝑦2)n⇒ 𝑥1 ≤ 𝑥2 or 𝑦1 ≤ 𝑦2n⇒ 𝑥2 ≤ 𝑥1 or 𝑦2 ≤ 𝑦1nThey may or may not be true. For example (1,2) and (3,4)n1 ≤ 3 and 2 ≤ 4 but 3 ¢ 1 and 4 ¢ 2.n∴ 𝑅 is not symmetric. For transitive :nTake pairs as (3,9), (4,6), (2,7)n(3,9)(4,6) as 4 ≥ 3n(4,6)(2,7) as 7 ≥ 6nBut (3,9)𝑅(2,7), as neither 2 ≥ 3 nor 7 ≥ 9nSo, 𝑅 is not transitive.n

Explanation

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