## inverse

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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: XY

(if it exists) is another function, f–1, such that f–1: YX

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 PQ. The statement QP.

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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.

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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.