Elementary Foundations: An introduction to topics in discrete mathematics

In $\mathbb{N}\text{,}$ $\mathbb{Z}\text{,}$ $\mathbb{Q}\text{,}$ or $\mathbb{R}\text{,}$ we use the usual $\le$ to describe when one number is (literally) less than or equal to another.

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 $\subseteq$ on $\mathcal{P}(U)$ acts very similarly to how $\le$ acts on a set of numbers.

Similar to Example 19.1.2, if $H$ and $H^{\prime }$ are subgraphs of a graph $G$ such that $H^{\prime }$ is a subgraph of $H\text{,}$ we might think of $H^{\prime }$ as being “less than or equal to” $H\text{.}$ That is, if we write $\mathcal{S}(G)$ to mean the set of all subgraphs of $G\text{,}$ then we can use the subgraph relation $⪯$ to describe when one subgraph of $G$ is “smaller than or equal to” another.