Prove or Disprove Let A B S and T be sets such that A is equ

Prove or Disprove. Let A, B, S, and T be sets such that A is equinumerous to B, S is equinumerous to T, A S, and B T. Then S \\ A is equinumerous to T \\ B.

Solution

Given that n(A) = n(B)

n(S) = n(T)

Since A is a subset of S, all elements of A are in S

Similarly B is a subset of T, hence all elements of B are in T

To check whether n(S\\A) = n(T\\B)

Let A = (1,3,5,7), B=(2,4,6,8)

S = (1,3,5,7,9) , T = (2,4,6,8,10,12)

satisfying all the given conditions

Now S\\A = (9) and n(S\\A) = 1

T\\B = (10,12) and n(T\\B) = 2

This disproves the given statement that

S\\A is equinumerous to T\\B