Proving Complement Laws

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The problem I am working on is:

Proof the following: $A∪ \bar{A}=U$

As with all proofs, I commenced this proof by using the definition of a union:

$A∪ \bar{A} = \{x|x \in A \vee x \in \bar{A}\}$

Using the definition of the complement of A:

$A∪ \bar{A} = \{x|x \in A \vee ( x \in U \wedge x \notin A) \}$

I then proceeded to use the distributive law, then the domination law, until I noticed a pattern--I was going in circles. Have I started my proof incorrectly?

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1 Answer

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$$A \cup \bar{A} = \{x|x \in A \vee ( x \in U \wedge x \notin A) \}$$

$$A \cup \bar{A} = \{x|(x \in A \vee x \in U) \land ( x \in A \lor \notin A) \}$$

$$\text{Note that}\;\; x \in A \lor x\notin A \iff x \in U.$$

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