Is the cardinality of set $A$ + $A^c$ greater than $A^c$?

Question

I'm working on a problem that involves the cardinality of a set and its complement. Basically, we have a set $A$ and its complement $A^c$ and I want to know if the conjecture $|A \cup A^c| > |A^c|$ or $|A \cup A^c| \geq |A^c|$ is true. A formal theorem or proof would be much appreciated.

Answer 1

Fix a universal set $X$ and a subset $A\subseteq X$. Since $A^c\subseteq A\cup A^c$ (in fact, $A\cup A^c=X$), we do indeed have $|A\cup A^c|\geq |A^c|$.

However, a strict inequality does not always hold. Indeed, consider $X:=\mathbb R$ and $A:=\mathbb R\setminus [0,1]$, so we have $|A\cup A^c|=|\mathbb R|=|[0,1]|=|A^c|$.