If A and B are the subsets of a universal set u then find A intersection B^c
Answers
f A is a subset of B, then any element x of A is element of B. This also means that all elements of A are elements of AUB. On the other hand, there are no elements of A that are not elements of B. This means that AUB = B.
This proves the first part.
We need to prove the opposite statement
2. AUB =B ⇒ A⊂B
Now, we know that the union of sets A and B is B. Any element that belongs to B, also belongs the union of sets A and B. So, the union of sets A and B contains exactly the elements of B. If x is element of B, then x belongs to B, and of the other side of the identity, it belongs either of B or A. So, for some of these x they are elements of A.
If an element y is NOT an element of B, then it is not an element of B and it is not an element of A.
So, all elements y that are not elements of B are not elements of AUB. We arrive at a situation, that all elements of A are elements of AUB, and all the elements not in A are not in B. This means that A is a subset of B.
Proving the two statements A⊂B ⇒ AUB =B and
AUB =B ⇒ A⊂B
means that A⊂B⇔A∪B=B.