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 \(\mathbb{R}\) is the inverse element of x for this operation
- The inverse of x for multiplication in \(\mathbb{R}\) is the inverse element of x for this operation.
- The reciprocal relationship \(f^{-1}\) of a function \(f\) defined in \(\mathbb{R}\) is the inverse element of \(f\) for the composition of functions, since \(f^{-1}\space ο\space f = I_{\mathbb{R}}\), where \( I_{\mathbb{R}}\) is the identity relation on \(\mathbb{R}\).