## Quick Reference

*R*. A derived relation *R*^{−1} such that

whenever *x R y*

then *y R*^{−1}*x*

where *x* and *y* are arbitrary elements of the set to which *R* applies. The inverse of “greater than” defined on integers is “less than”.

The inverse of a function *f*: *X* → *Y* (if it exists) is another function, *f*^{−1}, such that *f*^{−1}: *Y* → *X* and *f*(*x*) = *y* implies *f*^{−1}(*y*) = *x* It is not necessary that a function has an inverse function.

*f*: *X* → *Y*

*f*^{−1}: *Y* → *X*

*f*(*x*) = *y* implies *f*^{−1}(*y*) = *x*

Since for each monadic function *f* a relation *R* can be introduced such that *R* = {(*x,y*)} | *f*(*x*) = *y*} then the inverse relation can be defined as *R*^{−1} = {(*y,x*)} | *f*(*x*) = *y*} and this always exists. When *f*^{−1} exists (i.e. *R*^{−1} is itself a function) *f* is said to be **invertible** and *f*^{−1} is the inverse (or converse) **function**. Then, for all *x*, *f*^{−;1}(*f*(*x*) = *x*

*R* = {(*x,y*)} | *f*(*x*) = *y*}

*R*^{−1} = {(*y,x*)} | *f*(*x*) = *y*}

*f*^{−;1}(*f*(*x*) = *x*

To illustrate, if *f* is a function that maps each wife to her husband and *g* maps each husband to his wife, then *f* and *g* are inverses of one another.

*P*→*Q*. The statement *Q*→*P*.

**From:**
inverse
in
A Dictionary of Computing »

*Subjects:*
Computing.

## Related content in Oxford Index

##### Reference entries

Users without a subscription are not able to see the full content. Please, subscribe or login to access all content.