# Operations on sets

1. Mar 7, 2008

### chocolatelover

1. The problem statement, all variables and given/known data
Prove the following statement:

A is a subset of B if and only if (X/B) is a union of (X/A)

2. Relevant equations

3. The attempt at a solution

I really don't understand how to prove these types of problems. I was thinking about proving the contrapositive, which would be If A is not a subset of B, then (X/B) is not a union of (X/A), right?

Could someone please show me what to do?

Thank you very much

Last edited: Mar 7, 2008
2. Mar 7, 2008

### Mathdope

I don't quite understand what you mean when you say X/B is not a union of X/A. Can you clarify please?

3. Mar 7, 2008

### HallsofIvy

What do "(X/B) is a union of (X/A)" and "(X/B) is not a union of (X/A)" mean? I only know "union" as an operation on two sets.

Oh, and the contrapositive of "if A then B" is NOT "if not A then not B". That is the inverse and the truth of one does not imply the truth of the other. The contrapositive is "If not B then not A" and, here, that would be "If (X/B) is not a union of (X/A) then A is not a subset of B" though I still don't know what "union" means here!

Is it possible that you just meant to have "subset" again? "If X is a subset of B then (X\B) is a subset of (X\A)" is a true statement. (Notice also that I have reversed "/" to "\". "/" implies a division (which is not defined for sets) while "\" is the "set difference".

If that is true, that you want to prove "If X is a subset of B then (X\B) is a subset of (X\A)", I would not try to prove the contrapositive but prove it directly. The standard way to prove "P is a subset of Q" is to say "if x is a member of P" and prove, using whatever properties P and Q have, "therefore x is a member of q".

Here, you would start "if x is a member of (X\B), then x is a member of X but x is NOT a member of B" (using, of course, the definition of "X\B"). Now, what does that, together with the fact that A is a subset of B, tell you about whether or not x is a member of A?

4. Mar 7, 2008

### chocolatelover

Thank you very much

By "union" I ment that "X is such that x is an element of A or x is an element of B" Could you please show me what to do in this case?

Thank you