If R is a relation on Z (set of all integers) defined by xRy, iff |x − y| ≤ 1, then
(a) R is reflexive
(b) R is symmetric
(c) R is transitive
(d) R is not symmetric
(e) R is not transitive
Choose the most appropriate answer from the options given below
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (a), (b) and (e) only
Reflexive: for every integer x,
\( |x - x| = 0 \le 1 \)
Symmetric: if xRy, then
\( |y - x| = |x - y| \le 1 \)
Transitive fails, by a counterexample:
\( |1 - 2| = 1 \le 1,\ \text{so } 1R2 \)
\( |2 - 3| = 1 \le 1,\ \text{so } 2R3 \)
\( |1 - 3| = 2 \gt 1,\ \text{so } (1, 3) \notin R \)
So R is reflexive and symmetric, but not transitive.
Checking the statements
(a) R is reflexive — Correct. |x − x| = 0 ≤ 1 for every x.
(b) R is symmetric — Correct. |y − x| = |x − y|.
(c) R is transitive — Incorrect. 1R2 and 2R3, but |1 − 3| = 2 > 1.
(d) R is not symmetric — Incorrect. |y − x| = |x − y|, so R is symmetric.
(e) R is not transitive — Correct. 1R2 and 2R3, but 1 is not related to 3.