# Prove using sets

1. Feb 14, 2008

### Piffo

1. The problem statement, all variables and given/known data

Prove that A intersection B is a subset of A union B without using Venn diagrams

2. Relevant equations

No relevant equations really

3. The attempt at a solution

What i am thinking is this:

A union B = A+B- A intersection B shows the intersection is within the union
So A intersection B = -A union B +A+B, which tells me the intersection is smaller than the union.

Can anybody give me some feedback please?

2. Feb 14, 2008

### HallsofIvy

Staff Emeritus
What you should be thinking about is this:
A is a subset of B if and only if for all $x\in A$, $x\in B$, the definition of "subset".

Start with "if x is in A intersect B then---", use whatever properties x must have because of that and wind up with "therefore x is in A union B".

3. Feb 14, 2008

### PingPong

Exactly what HallsofIvy said. Any time that you want to show something is a subset, you must show that all of the elements of the subset are also in the larger set. This one is particularly simple, because you only have two steps in the logic.

The same procedure can be used to show that two sets are equal. For example, if you want to show that the set A is equal to the set B, first show that all elements of A are in B, and also that all elements of B are in A. Then you can conclude that A and B are equal because they have the same elements.