# Thread: Set theory proof help

1. ## Set theory proof help

I need to prove that for all sets A and B, if the complement of A is a subset of B then the union of A and B is equal to U, the universal set.

So far, I've come up with this:

Suppose the complement of A is a subset of B and the union of A and B is not equal to U. Let x be part of the union of A and B. By definition of union, x is a part of A or x is a part of B. Here is where I get stuck...what should I do next? Am I on the right track?

Thanks for any help.

2. Given that $A^c \subseteq B$.
It is always true that $A \cup B \subseteq U$.
So suppose that $x \in U$ if $x \in A$ then we are done.
If $x \notin A$ the by the given $x \in B$ and we are done.
Therefore $A \cup B = U$