How to Approach Solving Set Theory Equalities

[racshot65]

Hi,

I'm struggling to understand how to approach set theory equality questions for example:

True or false?

(A n B) is a subset of (A u B)


Is quite simple as its obvious the intersection will contain everything that is in the union

But what about a more complex question like ...


True or false

(A u B) n (C u D) = (A n (C u D)) u (B n (C u D))


There must be some method you follow to tackle a question like this as it isn't obvious like the previous question ?


My question is what is the method ?


Thanks

 

[Robert1986]

In general, if you have something like this:

Prove: $S_1 = S_2$ then you show that $S_1 \subset S_2$ and $S_2 \subset S_1$. Then the equality follows. In the above example, you would say, let $x \in (A \cup B) \cap (C \cup D)$ now you have to show that x is an element of the RHS of your equation. Then you "just" do this in the opposite direction. So, you know that $x$ is in either A or B AND it is in either C or D. Now, it should be clear (with a little work) that $(A\cup B) \cap (C \cup D) \subset (A \cap (C \cup D)) \cup (B \cap (C \cup D))$. Now just show the other direction.