Let R be the relation on the set N of natural numbers defined by \(R = \{(a, b) : 2a + b = 12\}\). Consider the following statements:
(A) R is reflexive.
(B) R has exactly 5 ordered pairs.
(C) R is symmetric.
(D) \((4, 4) \in R\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B) and (D) only
From \(2a + b = 12\):
\(b = 12 - 2a\)
b must be a natural number, so \(12 - 2a \ge 1\), which gives \(a = 1, 2, 3, 4, 5\):
\(R = \{(1, 10), (2, 8), (3, 6), (4, 4), (5, 2)\}\)
So R has 5 pairs, it contains (4, 4), it does not contain (1, 1), and it contains (1, 10) but not (10, 1).
Checking the statements
(A) R is reflexive. — Incorrect. \(2 \times 1 + 1 = 3 \ne 12\), so \((1, 1) \notin R\).
(B) R has exactly 5 ordered pairs. — Correct. \(a = 1, 2, 3, 4, 5\).
(C) R is symmetric. — Incorrect. \((1, 10) \in R\), but \(2 \times 10 + 1 = 21 \ne 12\), so \((10, 1) \notin R\).
(D) \((4, 4) \in R\) — Correct. \(2 \times 4 + 4 = 12\).