The relation R on the set \(\mathbf{R}\) of real numbers defined by \(R=\{(x, y): y=x+2\}\) is:
(A) Reflexive but not symmetric
(B) Neither reflexive nor symmetric
(C) Reflexive but not transitive
(D) Neither reflexive nor transitive
(E) Not reflexive but symmetric
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (B) and (D) only
A pair (x, y) is in R exactly when y = x + 2.
Reflexive needs (x, x) in R for every x, that is x = x + 2, which never holds:
\((1, 1) \notin R\)
Symmetric fails:
\((1, 3) \in R \text{ since } 3 = 1+2\)
\((3, 1) \notin R \text{ since } 1 \ne 3+2\)
Transitive fails:
\((1, 3) \in R,\ (3, 5) \in R\)
\((1, 5) \notin R \text{ since } 5 \ne 1+2\)
So R is not reflexive, not symmetric and not transitive.
Checking the statements
(A) Reflexive but not symmetric — Incorrect. R is not reflexive: (1, 1) ∉ R.
(B) Neither reflexive nor symmetric — Correct. (1, 1) ∉ R, and (1, 3) ∈ R but (3, 1) ∉ R.
(C) Reflexive but not transitive — Incorrect. R is not reflexive: (1, 1) ∉ R.
(D) Neither reflexive nor transitive — Correct. (1, 1) ∉ R, and (1, 3), (3, 5) ∈ R but (1, 5) ∉ R.
(E) Not reflexive but symmetric — Incorrect. R is not symmetric: (1, 3) ∈ R but (3, 1) ∉ R.