# 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 $$f$$.