This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 } ....
View
Full
Document
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.
 Spring '08
 howard
 Math

Click to edit the document details