inverse

If f:X->Y is a function then a function g:Y->X is an inverse of f iff fg=1_Y and gf=1_X. (Here 1_X,1_Y are the identity functions on X,Y.)

Notes

• If an inverse of f exists it is unique and we usually denote it by f^-1.
• f has an inverse iff it is bijective.
• The function f:R->R defined by f(x)=x+1 has an inverse defined by f^-1(x)=x-1 but the same rule on the positive real numbers gives an injective (but not bijective) function h:R₊->R₊ which is not invertible (essentially because x -> x-1 is not well defined as a function R₊->R₊).

In logic, there are four reformulations of any given implication: the implication itself, the converse, the inverse, and the contrapositive.

For a given implication a implies b, the inverse is formed by not a implies not b.

Note that an implication and its contrapositive are logically equivalent, while its converse and inverse are logically equivalent. Note further that the converse and inverse are usually logically unrelated to the implication and its contrapositive, except in the case of an iff.

An example: "If it's raining then the streets are wet". a is "it's raining" and b is "the streets are wet". Here's a truth table:

   a   |   b   | not a | not b | a --> b | not a --> not b | a <-> b
---------------------------------------------------------------------
   T   |   T   |   F   |   F   |    T    |        T        |    T
   T   |   F   |   F   |   T   |    F    |        T        |    F
   F   |   T   |   T   |   F   |    T    |        F        |    F
   F   |   F   |   T   |   T   |    T    |        T        |    T

The last column is "a if and only if b" which is true exactly when a --> b and b --> a, i.e. "a implies b and b implies a". For the iff statement, all four implication forms from above are logically equivalent.

In*verse" (?), a. [L. inversus, p. p. of invertere: cf. F. inverse. See Invert.]

1.

Opposite in order, relation, or effect; reversed; inverted; reciprocal; -- opposed to direct.

2. Bot.

Inverted; having a position or mode of attachment the reverse of that which is usual.

3. Math.

Opposite in nature and effect; -- said with reference to any two operations, which, when both are performed in succession upon any quantity, reproduce that quantity; as, multiplication is the inverse operation to division. The symbol of an inverse operation is the symbol of the direct operation with -1 as an index. Thus sin-1 x means the arc whose sine is x.

Inverse figures Geom., two figures, such that each point of either figure is inverse to a corresponding point in the order figure. -- Inverse points Geom., two points lying on a line drawn from the center of a fixed circle or sphere, and so related that the product of their distances from the center of the circle or sphere is equal to the square of the radius. -- Inverse, ∨ Reciprocal, ratio Math., the ratio of the reciprocals of two quantities. -- Inverse, ∨ Reciprocal, proportion, an equality between a direct ratio and a reciprocal ratio; thus, 4 : 2 : : , or 4 : 2 : : 3 : 6, inversely.

 

© Webster 1913.

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In"verse, n.

That which is inverse.

Thus the course of human study is the inverse of the course of things in nature. Tatham.

 

© Webster 1913.