Proof of A Intersection (A union B) = A for Sets A and B | Simple Set Proof

[iamalexalright]

Homework Statement

For any sets A and B, prove that

A\cap(A\cupB) = A



2. The attempt at a solution
Now keep in mind I don't have any experience with proofs(and I am looking for a nudge in the right direction not a full proof).

Here was my first instinct(and don't yell at me too much for it):
Suppose x \in A \cup B
Then x \in A or x \in B
If x \in A
then x \in A \cap A
so A = A
IF x \in B


Now after writing that I felt that this is not a good way to prove the problem(or a way to do it at all). So any hints would be appreciated.

 

[Dick]

The only cases you really need to worry about are i) x is in A and ii) x is not in A. Can you handle those two?

 

[statdad]

You need to show

<br /> A \cap \left( A \cup B \right) = A,<br />

correct? This means you must show that each set is a subset of the other.
1. Start with x \in A and show that it has to follow that

<br /> x \in A \cap \left(A \cup B \right)<br />

This will give that A \cap \left(A \cup B \right) \supseteq A

2. Now pick x \in A \cap \left(A \cup B\right). You need to show that this means x \in A (this should be rather easy). This will show that

<br /> A \cap \left(A \cup B \right) \subseteq A<br />

and you will be done.