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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.
From: inverse in A Dictionary of Computing »
Subjects: Computing.
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