Cardinality of Complement

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Theorem

Let $T \subseteq S$ such that $\left|{S}\right| = n, \left|{T}\right| = m$.


Then:

$\left|{\complement_S \left({T}\right)}\right| = \left|{S \setminus T}\right| = n - m$

where:


Proof

The result is obvious for $S = T$ or $T = \varnothing$.

Otherwise, $\left\{{T, S \setminus T}\right\}$ is a partition of $S$.

If $\left|{S \setminus T}\right| = p$, then by the Fundamental Principle of Counting, $m + p = n$ and the result follows.

$\blacksquare$


Sources