Identity relation

In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. The equality between A and B is written A = B, and pronounced A equals B. The symbol "=" is called an "equals sign". Thus there are three kinds of equality, which are formalized in different ways.

These may be thought of as the logical, set-theoretic and algebraic concepts of equality respectively.

The etymology of the word is from the Latin (“equal”, “like”, “comparable”, “similar”) from (“equal”, “level”, “fair”, “just”).

Leibniz characterized the notion of equality as follows:

In this law, "P(x) if and only if P(y)" can be weakened to "P(x) if P(y)"; the modified law is equivalent to the original.

Instead of considering Leibniz's law as a true statement that can be proven from the usual laws of logic (including axioms about equality such as symmetry, reflexivity and substitution), it can also be taken as the definition of equality. The property of being an equivalence relation, as well as the properties given below, can then be proved: they become theorems.

The substitution property states:

In first-order logic, this is a schema, since we can't quantify over expressions like F (which would be a functional predicate).

Some specific examples of this are:

The reflexive property states:

This property is generally used in mathematical proofs as an intermediate step.

The symmetric property states:

The transitive property states:

These three properties were originally included among the Peano axioms for natural numbers. Although the symmetric and transitive properties are often seen as fundamental, they can be proved if the substitution and reflexive properties are assumed instead.

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