# Homework Help: More set theory

1. Jul 29, 2006

### supasupa

The question is

If A U B is a subset of A intersect B, then prove that A=B

Now i can see this in my head and it makes sense that the elements in set A and Set B would have to be the same. The problem that i have with subset questions is how to prove that this is the case. I can start by saying

For all x ( x is an element of (A U B) --> x is an element of (A intersect B))
( x is an element of (A U B) --> x is an element of A AND B))
( x is an element of (A U B) --> x is an element of A AND x is an element of B))

Where do i go from here??? Any help would be great

2. Jul 29, 2006

### d_leet

I think that this approach will work, note that the intersection of any number of sets will always be a subset of any one of those sets, and for the union of any number of sets, any set in that union will be a subset of the union. So then show that based on what you know A is a subset of B and B is a subset of A, and they are consequently equal.

3. Jul 29, 2006

### 0rthodontist

This doesn't make sense--"x is an element of A AND B." What kind of set is "A AND B"? You should skip this step and go straight to saying x is an element of A and x is an element of B.

A simple way to approach the problem is to suppose that x is an element of A. Then you want to show that it is an element of B. The reverse direction is essentially the same. Here's how your proof could start:
"Let x be an element of A. Then x is an element of A or x is an element of B. So, x is an element of A u B."

4. Jul 29, 2006

### HallsofIvy

To prove "A= B", you have to do two things: prove "if x is an element of A, then it is an element of B" and prove "if x is an element of B, then it is an element of A".

You might start by saying "Let x be an element of A. Then it is an element of A union B. Since A union B is a subset of A intersect B, then ..."

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