# lec0323 - Functions A function is a special type of...

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Functions A function is a special type of relation. In particular, here are the rules for a relation to be a function, for an arbitrary relation R A x B : For each element in a A, we must have EXACTLY 1 element in R such that a is the first term of the ordered pair. In English that means for each element in the set A it MUST BE related to exactly one element in the set B. Note that this means we must have |A| |B|. Typically, we call the set A the domain and the set B the co-domain. It is possible that all of the possible values of f(a) (when a A) form only a proper subset of B. Thus the set of possible values of the function, which can be more formally written as follows: f(A) = { f(a) | a A } is known as the range of the function. Here is an example of a function : Let the set A = { the set of words in the English language } For all a A, define f(a) as follows: f(a) = the number of letters in the word a. Thus, we have that the domain is the set of all English words, the co-domain is the set of all positive integers, and the range is all positive integers less than 29, (assuming that antidisestablishmentarianism is the longest word in the English language).

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However, consider the following relation from a couple lectures ago: Cocktails = {(Orange Juice, Vodka), (Cranberry Juice, Vodka),
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lec0323 - Functions A function is a special type of...

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