Consider the following relations, where Z is the set of all integers and R is the set of real numbers.
Match List-I with List-II
| List-I | List-II | ||
| (A) | \(\{(T_1, T_2) : T_1 \text{ is congruent to } T_2\}\) on the set of all triangles in a plane | (I) | Reflexive and transitive but not symmetric |
| (B) | \(\{(a, b) : a - b \ge 3\}\) on Z | (II) | Symmetric but neither reflexive nor transitive |
| (C) | \(\{(a, b) : |a| \le |b|\}\) on R | (III) | An equivalence relation |
| (D) | \(\{(a, b) : |a - b| = 1\}\) on Z | (IV) | Transitive but neither reflexive nor symmetric |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
(A) \(\{(T_1, T_2) : T_1 \text{ is congruent to } T_2\}\) on the set of all triangles in a plane
This matches (III) An equivalence relation.
every triangle is congruent to itself, so reflexive
if \(T_1\) is congruent to \(T_2\), then \(T_2\) is congruent to \(T_1\), so symmetric
if \(T_1\) is congruent to \(T_2\) and \(T_2\) to \(T_3\), then \(T_1\) is congruent to \(T_3\), so transitive
(B) \(\{(a, b) : a - b \ge 3\}\) on Z
This matches (IV) Transitive but neither reflexive nor symmetric.
\(a - a = 0 \lt 3\), so not reflexive
\(5 - 0 = 5 \ge 3\) but \(0 - 5 = -5\), so (5, 0) is in it and (0, 5) is not: not symmetric
\(a - c = (a - b) + (b - c) \ge 3 + 3 = 6\), so transitive
(C) \(\{(a, b) : |a| \le |b|\}\) on R
This matches (I) Reflexive and transitive but not symmetric.
\(|a| \le |a|\), so reflexive
\(|1| \le |2|\) but \(|2| \le |1|\) is false, so not symmetric
\(|a| \le |b| \le |c|\) gives \(|a| \le |c|\), so transitive
(D) \(\{(a, b) : |a - b| = 1\}\) on Z
This matches (II) Symmetric but neither reflexive nor transitive.
\(|a - a| = 0 \ne 1\), so not reflexive
\(|b - a| = |a - b|\), so symmetric
(0, 1) and (1, 0) are in it, but \(|0 - 0| = 0\), so (0, 0) is not: not transitive