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$$\cup$$B) = 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

$$A \cap \left( A \cup B \right) = A,$$

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

$$x \in A \cap \left(A \cup B \right)$$

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

$$A \cap \left(A \cup B \right) \subseteq A$$

and you will be done.