Cardinality of Complement

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:

• $\complement_S \left({T}\right)$ denotes the complement of $T$ relative to $S$
• $S \setminus T$ denotes the difference between $S$ and $T$.

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

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 19$: Theorem $19.2$