Definition:Identity (Equation)

Definition

An identity is an equation which is true for all values attained by the variables it contains.

Symbol

The symbol $\equiv$ can be used to distinguish an identity from a conditional equation, but frequently (and usually on $\mathsf{Pr} \infty \mathsf{fWiki}$) the equals sign $=$ is used instead.

Examples

Square of Sum

The result Square of Sum:

$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$

is an example of an identity.

Difference of Two Squares

The result Difference of Two Squares:

$\forall x, y \in \R: x^2 - y^2 \equiv \paren {x + y} \paren {x - y}$

is an example of an identity.

Also see

• Results about identities in the context of equations can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): equation
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): identity
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): equation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): identity