Cardinality of Set Difference

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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:

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

$\blacksquare$