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functions- lect notes

functions- lect notes - Unit Fn Functions Section 1 Some...

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Unit Fn Functions Section 1: Some Basic Terminology Functions play a fundamental role in nearly all of mathematics. Combinatorics is no ex- ception. In this section we review the basic terminology and notation for functions. Per- mutations are special functions that arise in a variety of ways in combinatorics. Besides studying them for their own interest, we’ll see them as a central tool in other topic areas. Except for the real numbers R , rational numbers Q and integers Z , our sets are normally finite. The set of the first n positive integers, { 1 , 2 , . . . , n } will be denoted by n . Recall that | A | is the number of elements in the set A . When it is convenient to do so, we linearly order the elements of a set A . In that case we denote the ordering by a 1 , a 2 , . . . , a | A | or by ( a 1 , a 2 , . . . , a | A | ). Unless clearly stated otherwise, the ordering on a set of numbers is the numerical ordering. For example, the ordering on n is 1 , 2 , 3 , . . . , n . A review of the terminology concerning sets will be helpful. When we speak about sets, we usually have a “universal set” U in mind, to which the various sets of our discourse belong. Let U be a set and let A and B be subsets of U . The sets A B and A B are the intersection and union of A and B . The set A \ B or A B is the set difference of A and B ; that is, the set { x : x A, x negationslash∈ B } . The set U \ A or A c is the complement of A (relative to U ). The complement of A is also written A and A . The set A B = ( A \ B ) ( B \ A ) is symmetric difference of A and B ; that is, those x that are in exactly one of A and B . We have A B = ( A B ) \ ( A B ). • P ( A ) is the set of all subsets of A . (The notation for P ( A ) varies from author to author.) • P k ( A ) the set of all subsets of A of size (or cardinality) k . (The notation for P k ( A ) varies from author to author.) The Cartesian product A × B is the set of all ordered pairs built from A and B : A × B = { ( a, b ) | a A and b B } . We also call A × B the direct product of A and B . If A = B = R , the real numbers, then R × R , written R 2 , is frequently interpreted as coordinates of points in the plane. Two points are the same if and only if they have the same coordinates, which says the same thing as our definition, ( a, b ) = ( a , b ) if a = a and b = b . Recall that the direct product can be extended to any number of sets. How can R × R × R = R 3 be interpreted? Definition 1 (Function) If A and B are sets, a function from A to B is a rule that tells us how to find a unique b B for each a A . We write f : A B to indicate that f is a function from A to B . c circlecopyrt Edward A. Bender & S. Gill Williamson 2005. All rights reserved.
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Functions We call the set A the domain of f and the set B the range 1 or, equivalently, codomain of f . To specify a function completely you must give its domain, range and rule.
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