Definition:Substitution (Set Theory)

This page is about Substitution in the context of Set Theory. For other uses, see Substitution.

Definition

Let $S$ and $T$ be non-empty sets such that $T$ is not a subset of $S$.

Let:

$s \in S$

$t \in T \setminus S$

where $\setminus$ denotes set difference.

Let $S'$ be the set defined as:

$S' = \paren {S \setminus \set s} \cup \set t$

That is, $S'$ is the set obtained by removing $s$ and replacing it with $t$ which is not in $S$.

The operation of replacing $s$ with $t$ is known as substitution.