I am quite new to set theory, and I have tried proving the following set theory question. So, I was wondering if my proof is considered to be valid.
Here is the question: Let $A$ and $B$ be subsets of $U$. Using double set inclusion, prove that $$(A\cap B)^c = A^c \cup B^c$$
My proof:
Let $x \in (A \cap B)^c$
$\Leftrightarrow$
$x \notin (A \cap B)$ by definition of set complement
$\Leftrightarrow$
$\neg (x \in A \text{ and } x \in B))$
$\Leftrightarrow$ $x \notin A \text{ or } x \notin B)$ by Demorgan's Law
$\Leftrightarrow$
$x \in A^c \cup B^c$ by definition of set union
Therefore, I have shown that $(A\cap B)^c = A^c \cup B^c$, as was required. ▯