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. The equality between A and B is written A = B, and pronounced "A equals B".[1] The symbol "=" is called an "equals sign". Two objects that are not equal are said to be distinct.

For example:

• ${\displaystyle x=y}$ means that x and y denote the same object.[2]
• The identity ${\displaystyle (x+1)^{2}=x^{2}+2x+1}$ means that if x is any number, then the two expressions have the same value. This may also be interpreted as saying that the two sides of the equals sign represent the same function.
• ${\displaystyle \{x\mid P(x)\}=\{x\mid Q(x)\}}$ if and only if ${\displaystyle P(x)\Leftrightarrow Q(x).}$ This assertion, which uses set-builder notation, means that if the elements satisfying the property ${\displaystyle P(x)}$ are the same as the elements satisfying ${\displaystyle Q(x),}$ then the two uses of the set-builder notation define the same set. This property is often expressed as "two sets that have the same elements are equal." It is one of the usual axioms of set theory, called axiom of extensionality.[3]