Proposition 2.5.3.1.3 (01LG): Properties of Characteristic Relations

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Table of Contents
Part 1: Sets
Chapter 2: Constructions With Sets
Section 2.5: Characteristic Functions
Subsection 2.5.3: The Characteristic Relation of a Set
Proposition 2.5.3.1.3: Properties of Characteristic Relations (cite)

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Proposition 2.5.3.1.3 ▶ Properties of Characteristic Relations

Let $f\colon X\to Y$ be a function.

The Inclusion of Characteristic Relations Associated to a Function. Let $f\colon A\to B$ be a function. We have an inclusion¹

¹Note: This is the $0$-categorical version of Chapter 9: Preorders, Definition 9.5.4.1.1.

Proof of Proposition 2.5.3.1.3.

Item 1: The Inclusion of Characteristic Relations Associated to a Function

The inclusion $\chi _{B}\webleft (f\webleft (a\webright ),f\webleft (b\webright )\webright )\subset \chi _{A}\webleft (a,b\webright )$ is equivalent to the statement “if $a=b$, then $f\webleft (a\webright )=f\webleft (b\webright )$”, which is true.

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