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

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

These maxima are well-defined. Every subset of the finite set $V$ is finite and has cardinality at most $|V|$ (A subset of a finite set is finite, with $\lvert B\rvert \le \lvert A\rvert$, and equality holds if and only if $B = A$). Thus the sizes occurring in either display form a subset of $\{0,\ldots,|V|\}$ that contains $0$, since the empty subset is both independent and a clique. A short induction on $|V|$ 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 $1$.