19.1: Motivation
In many of the sets we encounter, there is some notion of elements being “less than or equal to” other elements in the set.
Example \(\PageIndex{1}\): Comparing numbers.
In \(\mathbb{N}\text{,}\) \(\mathbb{Z}\text{,}\) \(\mathbb{Q}\text{,}\) or \(\mathbb{R}\text{,}\) we use the usual \(\mathord{\le}\) to describe when one number is (literally) less than or equal to another.
Example \(\PageIndex{2}\): Subset relationship as a measure of relative size.
If \(A,B\) are subsets of a universal set \(U\) such that \(A\) is a subset of \(B\text{,}\) we might think of \(A\) as being “less than or equal to” \(B\text{.}\) The relation \(\mathord{\subseteq}\) on \(\mathscr{P}(U)\) acts very similarly to how \(\mathord{\le}\) acts on a set of numbers.
Warning
The idea of \(A \subseteq B\) expressing a “less than or equal to”-like relationship between \(A\) and \(B\) is very different from cardinality-based ideas of smaller/larger for sets. See also Example 19.2.4.
Example \(\PageIndex{3}\): Subgraph relationship as a measure of relative size.
Similar to Example \(\PageIndex{1}\), if \(H\) and \(H'\) are subgraphs of a graph \(G\) such that \(H'\) is a subgraph of \(H\text{,}\) we might think of \(H'\) as being “less than or equal to” \(H\text{.}\) That is, if we write \(S(G)\) to mean the set of all subgraphs of \(G\text{,}\) then we can use the subgraph relation \(\mathord{\preceq}\) to describe when one subgraph of \(G\) is “smaller than or equal to” another.