# Logical Proofs Regarding Sets and Subsets

1. Oct 23, 2011

### enkrypt0r

1. The problem statement, all variables and given/known data

The following is all the information needed:

2. Relevant equations

There are, of course, all the basic rules of logic and set identities to be considered.

3. The attempt at a solution

Not really sure how to attempt this one, to be honest. I know that (A ⊆ B) can be written with a quantifier and is equivalent to ∀x[xϵA ⇒ xϵB] but I'm really not sure how to apply this information properly.

Thanks, guys. :)

2. Oct 23, 2011

### kru_

If A is a subset of B, then we know that any element of A is also an element of B.

Now, what if we take an element of U, that is not an element of B. What does that mean? Could such an element be an element of A?

3. Oct 24, 2011

### HallsofIvy

Staff Emeritus
In general you prove set X is a subset of set Y by starting "if x is a member of X" and then using the definitions of X and Y, and any other information you are given, to conclude "x is a member of Y".

To prove $\overline{B}\subset\overline{A}$ start with "if x is a member of $\overline{B}$ then x is not in B. Since, by hypothesis, $A\subset B$ x is not in A. ...