inverse

1 (converse) of a binary relation R. A derived relation R⁻¹ such that

whenever x R y

then y R⁻¹x

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.

2 See group.

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