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\capB), then by definition, x\inA and x \inB. Thus we proved that (A\capB)\subseteqA.

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\capB...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\capB), then by definition, x\inA and x \inB. Thus we proved that (A\capB)\subseteqA.

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\capB...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.