Consider the following relations on the set R of real numbers:
(A) \(R_1 = \{(a, b) : a \le b + 2\}\)
(B) \(R_2 = \{(a, b) : ab \ge 0\}\)
(C) \(R_3 = \{(a, b) : b = 2a\}\)
(D) \(R_4 = \{(a, b) : a - b \gt 0\}\)
Which of the above relations are reflexive?
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (B) only
A relation on R is reflexive when (a, a) is in it for every real a. Putting \(b = a\) in each condition:
\(R_1\): \(a \le a + 2\), true for every a
\(R_2\): \(a \cdot a = a^{2} \ge 0\), true for every a
\(R_3\): \(a = 2a\), true only for \(a = 0\)
\(R_4\): \(a - a = 0 \gt 0\), never true
So \(R_1\) and \(R_2\) are reflexive, while \(R_3\) and \(R_4\) are not.
Checking the statements
(A) \(R_1 = \{(a, b) : a \le b + 2\}\) — Correct. \(a \le a + 2\) always.
(B) \(R_2 = \{(a, b) : ab \ge 0\}\) — Correct. \(a^{2} \ge 0\) always.
(C) \(R_3 = \{(a, b) : b = 2a\}\) — Incorrect. (1, 1) is not in \(R_3\), since \(1 \ne 2\).
(D) \(R_4 = \{(a, b) : a - b \gt 0\}\) — Incorrect. \(a - a = 0\) is not greater than 0.