Cardinality of Complement

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

Then $$\left|{\mathcal{C}_S \left({T}\right)}\right| = \left|{S - T}\right| = n - m$$.

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

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

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