Friday, June 17, 2011

Mathematical functions

A function is similar to a relation.  It is a set of ordered pairs (a, b) where a is an element of some set A and b is an element of some set B.  For each element in A, there is exactly one ordered pair with that element in the first position.  Given such a function f, we say that it is a function from A to B, which is written f:A->B.  We could also say that f:A->B is a mapping from A to B, in which case we would say that f maps from A into B.  This is written b = f(a), similar to how a relation is written aRba is mapped into b by f, and we say that b is the image of a under f.

Something worth noting is that because there is exactly one ordered pair per element in A, there is exactly one image of each element in A under f.  In other words, an element of A has exactly one image, no more, no less.  However, multiple elements of A can share the same image.  If each image is the image of only a single element of A, we say that f is one-to-one.  If all the elements of B are images of elements in A under f, we say that the function f maps from A onto B, rather than "into".

A composition of two functions f:A->B and g:B->C is the function fog:A->C which maps from A into C.  Such a function is written c = g(f(a)) where a is an element of A and c is an element of C.  This works exactly how you think it would: First b = f(a) is calculated, and the resulting element b is fed into g.

And lastly we have something called a "permutation".  A permutation is a function f:A->A where A is not the empty set and f is both one-to-one and maps from A onto A.  What you end up with is a function which rearranges the elements in A.

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