Equality (mathematics)

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. Equality between A and B is written A = B, and pronounced "A equals B". In this equality, A and B are the members of the equality and are distinguished by calling them left-hand side or left member, and right-hand side or right member. Two objects that are not equal are said to be distinct.

A formula such as ${\displaystyle x=y,}$ where x and y are any expressions, means that x and y denote or represent the same object.^[1] For example,

${\displaystyle 1.5=3/2,}$

are two notations for the same number. Similarly, using set builder notation,

${\displaystyle \{x\mid x\in \mathbb {Z} {\text{ and }}0<x\leq 3\}=\{1,2,3\},}$

since the two sets have the same elements. (This equality results from the axiom of extensionality that is often expressed as "two sets that have the same elements are equal".^[2])

The truth of an equality depends on an interpretation ot its members. In the above examples, the equalities are true it the members are interpreted as numbers or sets, but are false if the members are interpreted as expressions or sequences of symbols.

An identity, such as ${\displaystyle (x+1)^{2}=x^{2}+2x+1,}$ means that if x is replaced with any number, then the two expressions take the same value. This may also be interpreted as saying that the two sides of the equals sign represent the same function (equality of functions), or that the two expressions denote the same polynomial (equality of polynomials).^[3][4]