Cliques, independent sets, clique number and independence number

Let $G=(V,E)$ be a finite simple graph. A subset $S\subseteq V$ is independent, or stable, when no two distinct elements of $S$ are adjacent. A subset $K\subseteq V$ is a clique when every two distinct elements of $K$ are adjacent.

The independence number and clique number are

$\alpha (G):=\max \{\mathrm{\mid }S\mathrm{\mid }:S\subseteq V\text{ is independent}\},\omega (G):=\max \{\mathrm{\mid }K\mathrm{\mid }:K\subseteq V\text{ is a clique}\}\mathrm{.}$

These maxima are well-defined. Every subset of the finite set $V$ is finite and has cardinality at most $\mathrm{\mid }V\mathrm{\mid }$ (A subset of a finite set is finite, with $\mid B\mid \le \mid A\mid$, and equality holds if and only if $B=A$). Thus the sizes occurring in either display form a subset of $\{0,\dots ,\mathrm{\mid }V\mathrm{\mid }\}$ that contains $0$, since the empty subset is both independent and a clique. A short induction on $\mathrm{\mid }V\mathrm{\mid }$ shows that every such subset has a largest element: either its top element occurs, or it is a subset of the preceding finite ordinal and the induction hypothesis applies (The principle of mathematical induction). In particular, for the null graph one has $\alpha (G)=\omega (G)=0$; for every nonnull graph both values are at least $1$.