Let \(A = \{1, 2, 3, 4\}\) and \(R = \{(1, 2), (2, 3), (3, 4)\}\). The minimum number of ordered pairs that must be added to R so that it becomes both reflexive and transitive is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 7
Reflexive: the 4 pairs (1, 1), (2, 2), (3, 3), (4, 4) must be added.
Transitive: the chains in R force new pairs:
(1, 2), (2, 3) force (1, 3)
(2, 3), (3, 4) force (2, 4)
(1, 3), (3, 4) force (1, 4)
After adding these, the relation is
\(\{(a, b) : a \le b\}\) on A
which is reflexive and transitive, so nothing more is needed. No pair (a, b) with \(a \gt b\) is required, since the relation need not be symmetric.
pairs added \(= 4 + 3 = 7\)