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

From Intersection is Subset:

$S \cap T \subseteq S$

$S \cap T \subseteq T$

From Subset of Finite Set is Finite:

$S \cap T$ is finite.

We have:

\(\ds \card {S \setminus T}\)

\(=\)

\(\ds \card {S \setminus \paren {S \cap T} }\)

Set Difference with Intersection is Difference

\(\ds \)

\(=\)

\(\ds \card S - \card {S \cap T}\)

Cardinality of Set Difference with Subset

$\blacksquare$