A set \(S_0\) is a subset of a set \(S_1\) if every element of \(S_0\) is an element of \(S_1\), but not necessarily vice versa.
For example, the set \(A = \{1, 2, 3\}\) is a subset of both \(B = \{1, 2, 3\}\) and \(C = \{1, 2, 3, 4\}\).
If \(S_1\) definitely has elements not in \(S_0\) we write \(S_0 \subset S_1\), otherwise we write \(S_0 \subseteq S_1\), much as we do for inequalities with \(x_0 < x_1\) and \(x_0 \leqslant x_1\).
In the first case, \(S_0\) is known as a proper subset of \(S_1\).