lecnotes6_1 - Lecture Notes, Concepts of Mathematics...

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Unformatted text preview: Lecture Notes, Concepts of Mathematics (21-127) Lecture 1, Recitation AD, Spring 2008 6 Functions and Bijections 6.1 Functions READ: Textbook pages 125132. EXERCISES: Pages 153161, #117, 8185. 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 pre-image 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 pre-images of 9 . 6 has no pre-image 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 A : U R by A ( x ) = 1 if x A if x U \ A . Then 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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This note was uploaded on 01/25/2010 for the course MATH 21127 taught by Professor Howard during the Spring '08 term at Carnegie Mellon.

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lecnotes6_1 - Lecture Notes, Concepts of Mathematics...

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