Given relation \(R=\{(x, y): y=x+5, x \lt 4, x, y \in N\}\). Where N is a set of natural numbers then:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → R is transitive but neither reflexive nor symmetric.
The natural numbers less than 4 are 1, 2, 3, so listing the pairs:
\(R = \{(1, 6), (2, 7), (3, 8)\}\)
Reflexive: no. For example (1, 1) is not in R (no pair has y = x).
Symmetric: no. (1, 6) is in R but (6, 1) is not.
Transitive: yes. Transitivity needs a check only when (a, b) and (b, c) are both in R. The second entries 6, 7, 8 are never first entries, so no such pair of pairs exists and the condition is never broken (it holds vacuously).
So R is transitive but neither reflexive nor symmetric.
Tip: A relation with no chain (a, b), (b, c) is transitive by default; students often mark it “not transitive” because there is nothing to check.
NCERT (2026-27 edition):
Types of Relations
