justification of simple idea involving union

Question

How would you justify going from

$x\notin$ $A\cup B\ $

$x\notin A$ and $x\notin B$

in a proof, where the same values of x satisfy both statements.

would by definition of U be sufficient or would you have to explain it more?

Answer 1

I guess it depends on the context. If you want to prove it using formal logic, you need more. But in elementary set theory, you wouldn't need further justification, unless the exercise is to prove that.

Maybe you could write something like this: $A \cup B$ is, by definition, $\{ x : x\in A \lor x\in B\}$. Let $x \not\in A\cup B$ and suppose that $x \in A$. Then $x \in A \cup B$, which is absurd, so $x \in A$. Anagously, $x\not\in B$.