# inverse

**inverse** **1.** (**converse**) of a binary relation *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.

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*

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.

**2.** See group.

**3.** of a conditional *P*→*Q*. The statement *Q*→*P*.

# inverse

in·verse / ˈinvərs; inˈvərs/ •
adj. opposite or contrary in position, direction, order, or effect: *the well-observed inverse relationship between disability and social contact.* ∎ chiefly Math. produced from or related to something else by a process of inversion.•
n. [usu. in sing.] something that is the opposite or reverse of something else: *his approach is the inverse of most research on ethnic and racial groups.* ∎ Math. a reciprocal quantity, mathematical expression, geometric figure, etc., that is the result of inversion. ∎ Math. an element that, when combined with a given element in an operation, produces the identity element for that operation.DERIVATIVES: in·verse·ly adv.

# inverse

**inverse** adj. and sb. XVII. — L. *inversus*, pp. of *invertere*, f. IN-1 + *vertere* turn.

So **inversion** XVI. **invert** vb. XVI. — L. *invertere* ‘turn in, turn outside in’, reverse.

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