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# inverse

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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: 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.

f: XY

f−1: YX

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 PQ. The statement QP.

Subjects: Computing.

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