If R is a relation on \(A=\{a,b,c\}\) such that \(R=\{(a,a),(b,b)\}\), which element/elements should be included to make R an equivalence relation.
A. (c, c)
B. (c, c), (a, c), (c, a)
C. (a, b), (b, c), (a, c)
D. (b, c), (c, c), (c, a), (b, a)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A and B only
An equivalence relation on A is reflexive, symmetric and transitive. R already has (a, a) and (b, b).
So R must gain (c, c) to be reflexive, every added pair must come with its reverse pair, and whenever (x, y) and (y, z) are in R, (x, z) must be in R too.
Checking the statements
(A) (c, c) — Correct. R becomes {(a, a), (b, b), (c, c)}, the identity relation, which is an equivalence relation.
(B) (c, c), (a, c), (c, a) — Correct. Reflexive, each pair has its reverse, and (a, c), (c, a) give only (a, a) and (c, c), which are in R.
(C) (a, b), (b, c), (a, c) — Incorrect. (c, c) is missing, so R is not reflexive.
(D) (b, c), (c, c), (c, a), (b, a) — Incorrect. (b, a) is added without (a, b), so R is not symmetric.