They have different meanings in mathematics. As Mike says,
equality (represented by “=”) means that two things are the same.
For example, 3 = 3 = 3.0 = 1+2 …
and an infinite variety of other ways of expressing the same number,
but 3 does not equal any other number.
Equivalence is much more loosely defined.
An equivalence relationship (typically represented by “≅”)
is any relationship that satisfies the following properties:
- Reflexive: For any object X, it is true that X ≅ X.
- Symmetric: For any X and Y, if X ≅ Y, then Y ≅ X.
- Transitive: For any X, Y, and Z, if X ≅ Y and Y ≅ Z,
then X ≅ Z.
Clearly equality satisfies the above properties,
so equality is an equivalence relationship.
(In other words, equality is a subset (a special case) of equivalence.)
But equivalence relationships can be more interesting.
One of the best known equivalence relationships is modulo.
For example, in the modulo 10 equivalence relationship, 3 ≅ 13 ≅ 23 ≅ 33 …