Equality (mathematics) - Relation With Equivalence and Isomorphism

Relation With Equivalence and Isomorphism

See also: Equivalence relation and Isomorphism

In some contexts, equality is sharply distinguished from equivalence or isomorphism. For example, one may distinguish fractions from rational numbers, the latter being equivalence classes of fractions: the fractions and are distinct as fractions, as different strings of symbols, but they "represent" the same rational number, the same point on a number line. This distinction gives rise to the notion of a quotient set.

Similarly, the sets

and

are not equal sets – the first consists of letters, while the second consists of numbers – but they are both sets of three elements, and thus isomorphic, meaning that there is a bijection between them, for example

However, there are other choices of isomorphism, such as

and these sets cannot be identified without making such a choice – any statement that identifies them "depends on choice of identification". This distinction, between equality and isomorphism, is of fundamental importance in category theory, and is one motivation for the development of category theory.