inverse

Logic

An inverse statement is a statement derived from a conditional by negating both the antecedent and the consequent. Thus the inverse of the conditional statement ‘if \(A\) then \(B\)’ is ‘if not \(A\) then not \(B\).’ In symbolic logic this is written as \(\neg A \rightarrow \neg B\).

A conditional and its inverse are not logically equivalent, but the inverse and converse of a conditional are contrapositives of one another, and hence equivalent.

Analysis/Calculus/Precalculus/Topology

If (and only if) a function \(f\) is one-to-one, the inverse function \(f^{-1}\) exists and is defined by \(y = f(x)\) if and only if \(x = f^{-1}(y)\). (Compare converse relation.)

If \(f\) is any function with domain \(A\) and range \(B\), and if \(B^{\prime}\) is a subset of \(B\), then the inverse image of \(B^{\prime}\) in \(A\) is the set consisting of just those elements of \(A\) that are mapped to elements of \(B^{\prime}\) by \(f\).