18.2: Basics and Examples
What properties should a relation on a set have to be useful as a notion of “equivalence”?
- Each object in the set should be equivalent to itself. So the relation should be reflexive .
- Equivalence should be bidirectional. That is, a pair of equivalent objects should be equivalent to each other. So the relation should be symmetric .
- We should be able to infer equivalence from chains of equivalence. So the relation should be transitive .
Definition: Equivalence Relation
a relation on a set that is reflexive , symmetric , and transitive
Definition: \(\mathord{\equiv}\)
symbol for an abstract equivalence relation (instead of the letter \(R\) that we've been using for abstract relations up until now)
Example \(\PageIndex{1}\)
Let \(\mathscr{L}\) be the set of all possible logical statements built out of the statement variables \(p_1, p_2, p_3, \ldots\text{.}\) Show that logical equivalence of statements is an equivalence relation on \(\mathscr{L}\text{.}\)
Solution
Reflexive. We have \(A \Leftrightarrow A\) for every statement \(A\text{,}\) since \(A\) has the same truth table as itself.
Symmetric. If \(A \Leftrightarrow B\text{,}\) then \(A,B\) have the same truth table, so \(B \Leftrightarrow A\text{.}\)
Transitive. If \(A \Leftrightarrow B\) and \(B \Leftrightarrow C\text{,}\) then \(A\) has the same truth table as \(B\text{,}\) which has the same truth table as \(C\text{.}\) So \(A\) has the same truth table as \(C\text{,}\) i.e. \(A \Leftrightarrow C\text{.}\)
Definition: Equivalence Modulo \(n\)
an equivalence of integers, where two integers are equivalent if they have the same remainder when divided by \(n\)
Definition: \(m_1 \equiv_n m_2\)
integers \(m_1,m_2\) are equivalent modulo \(n\)
Verify that equivalence modulo \(n\) is an equivalence relation.