# Simple Set Theory - Union, Intersect and Complement

1. Jul 24, 2011

### BrianMath

1. The problem statement, all variables and given/known data
If A, B, and C are subsets of the set S, show that
$$A^C \cup B^C = \left(A \cap B\right)^C$$

2. Relevant equations
$$A^C = \{x \in S: x \not \in A\}$$
$$A\cup B = \{x \in S:\; x \in A\; or\; x\in B\}$$
$$A\cap B = \{x \in S:\; x \in A\; and\; x\in B\}$$

3. The attempt at a solution

$$A^C \cup B^C = \{x\in S:\; x\not \in A\; or\; x \not \in B\}$$
$$\left(A \cap B\right)^C = \{x\in S:\; x \not \in \left(A\cap B)\right \}$$
Since $\lnot A \lor \lnot B = \lnot \left(A \land B\right)$, we have that $\{x\in S:\; x\not \in A\; or\; x \not \in B\} = \{x\in S:\; x \not \in \left(A\cap B)\right \}$
$$\;\;\; \therefore A^C \cup B^C = \left(A \cap B\right)^C$$

Is it valid for me to relate the set operations to the analogous logic symbols to show that this is true? Such as, complement being the same as $\lnot$ (not) or the union and intersection being the same as $\lor$ (or) and $\land$ (and), respectively?

I'm sorry if this is obvious. I'm teaching myself Analysis out of Maxwell Rosenlicht's "Introduction to Analysis" text, so I want to make sure I'm not making any false assumptions or developing a false intuition of set theory.

2. Jul 24, 2011

### ArcanaNoir

It's okay to use "not" when dealing with the complement. It's also okay to use the equivalence of one thing to another, like -A or -B = -(A and B) as long as you know it's true, and you are sure to handle order of operations in these situations properly. The only time I can think of where it may be not okay to use such equivalences would be on a test, where the whole point would be showing the equivalence.