The relation R on the set N of natural numbers defined as \( \mathrm{R}=\{(x,y):y=x+3\ \text{and}\ x\le 5\} \) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → neither reflexive, nor symmetric nor transitive
With x a natural number and x ≤ 5, R has only five pairs:
\( \displaystyle \mathrm{R}=\{(1,4),\ (2,5),\ (3,6),\ (4,7),\ (5,8)\} \)
Reflexive: (1, 1) ∉ R (in fact no (a, a) is in R, since y = x + 3 ≠ x). So R is not reflexive.
Symmetric: (1, 4) ∈ R but (4, 1) ∉ R. So R is not symmetric.
Transitive: (1, 4) ∈ R and (4, 7) ∈ R, but (1, 7) ∉ R (7 ≠ 1 + 3). So R is not transitive.
Hence R is neither reflexive, nor symmetric nor transitive.