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.
The way you prove " " 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, " " is defined as the set of all x such that x is in A and x is in B. If then . Now, you need to show "if then ". So, what is the definition of " "?