19.1: Motivation

Elementary Foundations: An Introduction to Topics in Discrete Mathematics · LibreTexts

0%

Contents

    Text

    Size
    100%
    Theme
    Typeface
    Line length
    Spacing

    From LibreTexts, under GNU Free Documentation, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    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.

    This page titled 19.1: Motivation was last modified on Tue, 01 Sep 2026 03:36:00 GMT and is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Jeremy Sylvestre via source content that was edited to the style and standards of the LibreTexts platform.