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Unformatted text preview: Lecture Notes, Concepts of Mathematics (21127) Lecture 1, Recitation A–D, Spring 2008 6 Functions and Bijections 6.1 Functions READ: Textbook pages 125–132. EXERCISES: Pages 153–161, #1–17, 81–85. Recall from your previous mathematical knowledge that a function assigns exactly one output to each input. This is a special type of relation: Definition 6.1 Let A and B be sets. A function f from A to B (or, from A into B ), de noted f : A → B , is a relation from A to B that satisfies (i) Dom ( f ) = A (ii) If ( x,y ) ∈ f and ( x,z ) ∈ f , then y = z . In the case where A = B , we say f is a function on A . The function f is also called a mapping . We also refer to B as the codomain of f . Note that Rng( f ) is a subset of the codomain of f . The element y = f ( x ) ∈ B is called the value of f at x ∈ A or the image of x under f , y is the depenedent variable . x is a preimage of y under f and is the independent variable or argument of f . We may also consider the action of f on a set, i.e., f ( A ) = Rng( f ). Note: To verify that a given relation f from A to B is a function from A to B , it must be shown that every element of A appears as a first coordinate of exactly one ordered pair in f . The fact that each a ∈ A is used at least once as a first coordinate makes Dom( f ) = A ; the fact that a is used only once fulfills condition (ii) of the definition. Example 6.1 Let f be a function defined on Z by y = f ( x ) = x 2 . Then the domain of f is Z , the codomain of f is Z , and Rng ( f ) = { , 1 , 4 , 9 , 16 , 25 ,... } . The image of 4 is 16 , and both 3 and 3 are preimages of 9 . 6 has no preimage in the domain. Example 6.2 The empty set ∅ is a function on itself. In fact, if f : A → B and any one of f , A , or Rng ( f ) are empty, then all three are empty. Example 6.3 The identity relation I A on a set A is a function. We have I A ( x ) = x for all x ∈ A . 1 Example 6.4 Let X be the set of all people who have ever lived. Then m : X → X defined by y = m ( x ) iff “ y is the mother of x ” is a function (of course, disregarding biblical stories and feats of genetic engineering). Example 6.5 Assume that a universe U has been specified, and that A ⊆ U . Define χ A : U → R by χ A ( x ) = 1 if x ∈ A if x ∈ U \ A . Then χ A is a function on U and is commonly referred to as the characteristic function of A . Definition 6.2 Two functions f and g are equal if they have the same domains, the same codomains and, for every x in the domain, f ( x ) = g ( x ) . Note that the two functions f ( x ) = ( x 2 1) / ( x 1) and g ( x ) = x + 1 are not equal, as (1 , 2) ∈ g but (1 , 2) / ∈ f . 6.2 The Graph of a Function Definition 6.3 The graph of the function f : X → Y is the subset of X × Y consisting of pairs ( x,f ( x )) for all x ∈ X . That is, the graph of f is the set { ( x,f ( x )) ∈ X × Y  x ∈ X } ....
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 Spring '08
 howard
 Math, Inverse function, codomain, Bijections, cardinals

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