Inverse Element
The inverse element of an element x from a set E for an operation ⊕ defined on E is the element x' of E such that x ⊕ x' = n where n ∈ E is the identity element for the operation ⊕.
Examples
- The additive inverse of x for addition in [latex]\mathbb{R}[/latex] is the inverse element of x for this operation
- The inverse of x for multiplication in [latex]\mathbb{R}[/latex] is the inverse element of x for this operation.
- The reciprocal relationship [latex]f^{-1}[/latex] of a function [latex]f[/latex] defined in [latex]\mathbb{R}[/latex] is the inverse element of [latex]f[/latex] for the composition of functions, since [latex]f^{-1}\space ο\space f = I_{\mathbb{R}}[/latex], where [latex] I_{\mathbb{R}}[/latex] is the identity relation on [latex]\mathbb{R}[/latex].