Example 19.1.1. Comparing numbers.
In \(\N\text{,}\) \(\Z\text{,}\) \(\Q\text{,}\) or \(\R\text{,}\) we use the usual \(\mathord{\le}\) to describe when one number is (literally) less than or equal to another.
Section 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 19.1.2. 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 \(\powset{U}\) acts very similarly to how \(\mathord{\le}\) acts on a set of numbers.
Warning 19.1.3.
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 19.2.5.
Example 19.1.4. Subgraph relationship as a measure of relative size.
Similar to Example 19.1.2, 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 \(\subgraphset{G}\) to mean the set of all subgraphs of \(G\text{,}\) then we can use the subgraph relation \(\mathord{\partorder}\) to describe when one subgraph of \(G\) is “smaller than or equal to” another.
