Advanced Calc. proof, about sets and intersection.

[emira]

Homework Statement

Prove that: A is a subset of B if and only if (A intersection B)=A

Homework Equations

The Attempt at a Solution

I tried proving the right side, that is

(A $$\cap$$ B)=A
For two sets to be equal then they have to be subsets of each other...so:

(A $$\cap$$ B) $$\subseteq$$ A and A $$\subseteq$$ (A $$\cap$$ B)
So if we assume an element x $$\in$$ (A$$\cap$$B), then by definition, x$$\in$$A and x $$\in$$B. Thus we proved that (A$$\cap$$B)$$\subseteq$$A.

In not quite sure how to prove the opposite, because if x is an element of A, that doenst necessarily mean that x is an element of A$$\cap$$B...so i need help with the rest of it..or if you got any other ideas on how to approach it.

Thank you,
Emira!

 

[HallsofIvy]

Homework Statement

Prove that: A is a subset of B if and only if (A intersection B)=A

Homework Equations

The Attempt at a Solution

I tried proving the right side, that is

(A $$\cap$$ B)=A
For two sets to be equal then they have to be subsets of each other...so:

(A $$\cap$$ B) $$\subseteq$$ A and A $$\subseteq$$ (A $$\cap$$ B)
So if we assume an element x $$\in$$ (A$$\cap$$B), then by definition, x$$\in$$A and x $$\in$$B. Thus we proved that (A$$\cap$$B)$$\subseteq$$A.

Very good. That is exactly right!

In not quite sure how to prove the opposite, because if x is an element of A, that doenst necessarily mean that x is an element of A$$\cap$$B...so i need help with the rest of it..or if you got any other ideas on how to approach it.

Thank you,
Emira![/QUOTE]
For the opposite, notice that you haven't used the hypothesis that A is a subset of B. If x is in A, then, because A is a subset of B it is also in B. Since it is in both A and B, it is in $A\cap B$
Now you have to prove the implication the other way: If $A\cap B\subseteq A$ then $A\subseteq B$.