Let a and b be subsets of a universal set U Then A U B A if

Let a and b be subsets of a universal set U. Then A U B = A if and only if B is a subset of A.
Please explain

Solution

Solution:

We need to prove two things:
1) If A U B = A, then B is a subset of A
2) If B is a subset of A, then A U B = A

i) Suppose A U B = A. To prove that B is a subset of A, it is enough to show that every element of B is an element of A.

If x is an element of B, then clearly x is an element of A or x is an element of B, which implies that x is an element of A U B. Then since A U B = A, x is an element of A.

Therefore, B is a subset of A.

ii) Suppose B is a subset of A. To prove that A U B = A, we need to prove that every element in A U B is an element of A, and also vice versa.

If x is an element of A U B, then x is an element of A or x is an element of B. In the case that x is an element of A, then of course we have shown that x is an element of A.

In the case that x is an element of B, x is still an element of A because B is a subset of A. So in either case, x is an element of A.

If x is an element of A, then clearly x is an element of A or x is an element of B, which implies that x is an element of A U B.

Therefore, A U B = A.

From i) and ii) together, A U B = A if and only if B is a subset of A.