Cardinality of Set Difference

Theorem

Let $S$ and $T$ be sets such that $T$ is finite.

Then:

$\card {S \setminus T} = \card S - \card {S \cap T}$

where $\card S$ denotes the cardinality of $S$.

Proof

$S \cap T \subseteq S$
$S \cap T \subseteq T$
$S \cap T$ is finite.

We have:

 $\displaystyle \card {S \setminus T}$ $=$ $\displaystyle \card {S \setminus \paren {S \cap T} }$ Set Difference with Intersection is Difference $\displaystyle$ $=$ $\displaystyle \card S - \card {S \cap T}$ Cardinality of Set Difference with Subset

$\blacksquare$