Let \(R_0\) be the set of all non-zero real numbers, and let R be the relation on \(R_0\) defined by \(R = \{(a, b) : ab = 4\}\). Consider the following statements:
(A) \((2, 2) \in R\)
(B) R is reflexive.
(C) R is symmetric.
(D) R is transitive.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (C) only
Checking the pairs needed for each property:
\(2 \times 2 = 4\), so \((2, 2) \in R\)
\(1 \times 1 = 1 \ne 4\), so \((1, 1) \notin R\)
If \(ab = 4\), then \(ba = 4\), so R is symmetric.
\(1 \times 4 = 4\), so \((1, 4) \in R\)
\(4 \times 1 = 4\), so \((4, 1) \in R\)
but \((1, 1) \notin R\)
So (2, 2) is in R, and R is symmetric but neither reflexive nor transitive.
Checking the statements
(A) \((2, 2) \in R\) — Correct. \(2 \times 2 = 4\).
(B) R is reflexive. — Incorrect. \((1, 1) \notin R\), since \(1 \times 1 = 1\).
(C) R is symmetric. — Correct. \(ab = 4\) gives \(ba = 4\).
(D) R is transitive. — Incorrect. (1, 4) and (4, 1) are in R, but (1, 1) is not.