Given the relation \( R=\{(1,2),(2,3)\} \) on a set \( A=\{1,2,3\} \), then which of following statements are TRUE?
A. Minimum number of ordered pairs are added to R so that enlarged relation is reflexive is 3
B. Minimum number of ordered pairs are added to R so that enlarged relation is symmetric is 2
C. Minimum number of ordered pairs are added to R so that enlarged relation is an equivalence relation is 7
D. Minimum number of ordered pair are added to R so that enlarged relation is an equivalence relation is 6.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A, B and C only
A reflexive relation on A must contain (1, 1), (2, 2), (3, 3); none of them is in R, so 3 pairs are needed.
A symmetric relation needs (2, 1) with (1, 2), and (3, 2) with (2, 3), so 2 pairs are needed.
For an equivalence relation, besides these 3 + 2 pairs, transitivity needs (1, 3) from (1, 2) and (2, 3), and then symmetry needs (3, 1):
\( \displaystyle 3+2+2=7 \)
The enlarged relation is then all of A × A (9 pairs = 2 given + 7 added).
So the minimum numbers of pairs to be added are 3 (reflexive), 2 (symmetric) and 7 (equivalence relation).
Checking the statements
(A) Minimum number of ordered pairs are added to R so that enlarged relation is reflexive is 3 — Correct. (1, 1), (2, 2), (3, 3) must be added.
(B) Minimum number of ordered pairs are added to R so that enlarged relation is symmetric is 2 — Correct. (2, 1) and (3, 2) must be added.
(C) Minimum number of ordered pairs are added to R so that enlarged relation is an equivalence relation is 7 — Correct. The relation must become all 9 pairs of A × A, and R has 2.
(D) Minimum number of ordered pair are added to R so that enlarged relation is an equivalence relation is 6. — Incorrect. The number is 7, not 6.