Consider the following relations on the set Z of all integers.
Match List-I with List-II
| List-I | List-II | ||
| (A) | \(R_1 = \{(a, b) : a = b\}\) | (I) | Symmetric but neither reflexive nor transitive |
| (B) | \(R_2 = \{(a, b) : a \lt b\}\) | (II) | Reflexive but neither symmetric nor transitive |
| (C) | \(R_3 = \{(a, b) : a \ne b\}\) | (III) | An equivalence relation |
| (D) | \(R_4 = \{(a, b) : a \le b + 1\}\) | (IV) | Transitive but neither reflexive nor symmetric |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
(A) \(R_1 = \{(a, b) : a = b\}\)
This matches (III) An equivalence relation.
\(a = a\) always; \(a = b\) gives \(b = a\); \(a = b\) and \(b = c\) give \(a = c\).
(B) \(R_2 = \{(a, b) : a \lt b\}\)
This matches (IV) Transitive but neither reflexive nor symmetric.
\(1 \lt 1\) is false, so not reflexive
\(1 \lt 2\) but \(2 \lt 1\) is false, so not symmetric
\(a \lt b\) and \(b \lt c\) give \(a \lt c\), so transitive
(C) \(R_3 = \{(a, b) : a \ne b\}\)
This matches (I) Symmetric but neither reflexive nor transitive.
\(1 \ne 1\) is false, so not reflexive
\(a \ne b\) gives \(b \ne a\), so symmetric
(1, 2) and (2, 1) are in \(R_3\), but (1, 1) is not, so not transitive
(D) \(R_4 = \{(a, b) : a \le b + 1\}\)
This matches (II) Reflexive but neither symmetric nor transitive.
\(a \le a + 1\) always, so reflexive
\(0 \le 5 + 1\) but \(5 \le 0 + 1\) is false, so not symmetric
\(2 \le 1 + 1\) and \(1 \le 0 + 1\), but \(2 \le 0 + 1\) is false, so not transitive