Equality (mathematics)

In mathematics, equality is a relationship between two quantities or expressions, stating that they have the same value, or represent the same mathematical object.^[1] Equality between A and B is written A = B, and pronounced "A equals B". In this equality, A and B are distinguished by calling them left-hand side (LHS), and right-hand side (RHS). 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.^[2] 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".^[3])

The truth of an equality depends on an interpretation of its members. In the above examples, the equalities are true if 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).^[4][5]