Identity element
High school
Consider an operation that combines two things, like addition or multiplication. An identity element for the operation is one that leaves everything unchanged.
Undergraduate
Consider a set $S$ with a binary operation $\ast$. An identity element for $\ast$ is an element $e$ such that
\[e \ast a = a \ast e = a \quad \text{for all } a \in S.\]For addition, $e$ is called the additive identity and is usually written as $0$. For multiplication, $e$ is the multiplicative identity and is usually written $1$.
Examples
Examples
-
$(0, 0) \in \reals^2$, for vector addition
High school
$(0, 0)$ is the identity element for vector addition in $\reals^2$, because adding it changes nothing.
-
$x \mapsto x$, for composition of functions
High school
$x \mapsto x$ is the identity element for composition, because composing every function with it gives that function.
Properties
- If an operation has an identity element, it is unique: if $e$ and $e^{\prime}$ are both identities, then $e = e \ast e^{\prime} = e^{\prime}$.
Notes
The identity element of an operation is different from the identity function $x \mapsto x$, though the identity function is the identity element for composition of functions.