Let A= {1,2,3,4}. Let R be the equivalence relation on A × A defined by (a,b)R(c,d) iff a + d = b + c . Find the equivalence class [(1,3)].

Question

Philoid · Accepted Answer

The equivalence class [(1,3)] is the set {(c,d) : (1,3)R(c,d)}

Now as it is given that (a,b)R(c,d) iff a + d = b + c

Hence (1,3) R(c,d) iff 1 + d = 3 + c

⇒ (1,3) R(c,d) iff d – c = 2

Now as the set A contains {1, 2, 3, 4} hence we have only two pairs of (c, d) such that d – c = 2 which are (1, 3) and (2, 4)

Hence (1, 3) R (1, 3) and (1, 3)R(2, 4)

Hence the equivalence class [(1,3)] is the set {(1,3), (2,4)}