Proving A=B When A U B is a Subset of A Intersect B: Set Theory Explained

[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

 

[d_leet]

I think that this approach will work, note that the intersection of any number of sets will always be a subset of anyone 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.

 

[0rthodontist]

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

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."

 

[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 ..."