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relations and functions

relations and functions - Relations and Functions LING 178...

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9/4/2007 1 Relations and Functions LING 178 Relations A RELATION holds between two or more ordered entities. 2 is greater than 1 aardvark comes before anteater Homer is the father of Bart everyone admires Madonna Ordering is important. We cannot change the order without changing the meaning. Relations and ordering A set of two elements is a PAIR. The elements are not ordered. {1, 2} = {2,1} A set of two ordered elements is an ORDERED PAIR and is used to represent a relation. <1,2> <2,1> When there is a relation between one object and several other different objects, we list the relationships as separate pairs. <Homer, Bart> <Homer, Lisa> <Homer, Maggie>
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9/4/2007 2 Cartesian Product The CARTESIAN PRODUCT of two sets, A and B, is the set consisting of all ordered pairs formed by combining one element of A with one element of B. A x B = { <x,y> | x A and y B } A = {a, b, c} B = {1,2} what is A x B? a b c 1 2 A x B = { <a,1>, <a,2>, <b,1>, <b,2>, <c,1>, <c,2>} Is the following true? A x B = B x A What is B x B? Functions When we relate two sets of elements in such a way that every element of set A is paired with just one element of set B, we have a FUNCTION. Formally, a relation R from A to B is a function iff Each element in the domain [A] is paired with just one element in the range [B].
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