# Thread: Proving Intersections and Unions

1. ## Proving Intersections and Unions

(A intersect B) is contained in A is contained in (A union B)

I know I have to show containment both ways, meaning
(A intersect B) is contained in A
A is contained in (A union B)

Can you help me get started?

2. ## Re: Proving Intersections and Unions

To prove that X is contained in Y, show that every element of X is an element of Y. Use the definitions of union and intersection.

3. ## Re: Proving Intersections and Unions

Originally Posted by lovesmath
(A intersect B) is contained in A is contained in (A union B)

I know I have to show containment both ways, meaning
(A intersect B) is contained in A
A is contained in (A union B)
Can you help me get started?
If $\displaystyle x\in A\cap B$ then $\displaystyle x\in A~\&~x\in B$ therefore $\displaystyle x\in A$. That means $\displaystyle A\cap B\subseteq A.$

4. ## Re: Proving Intersections and Unions

The way you prove "$\displaystyle X\subseteq Y$" is to start "if x is in X" then use the definitions of X and Y to show "then x is in Y". As Plato said, "$\displaystyle A\cap B$" is defined as the set of all x such that x is in A and x is in B. If $\displaystyle x\in A\cap B$ then $\displaystyle x\in A$. Now, you need to show "if $\displaystyle x\in A$ then $\displaystyle x\in A\cup B$". So, what is the definition of "$\displaystyle A \cup B$"?