cal3 - 3.1 Chapter Three Vector Functions 3.1 Relations and...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 3.1 Chapter Three Vector Functions 3.1 Relations and Functions We begin with a review of the idea of a function. Suppose A and B are sets. The Cartesian product A B × of these sets is the collection of all ordered pairs ( , ) a b such that a A ∈ and b B ∈ . A relation R is simply a subset of A B × . The domain of R is the set dom R = { :( , ) } a A a b R ∈ ∈ . In case A = B and the domain of R is all of A , we call R a relation on A . A relation R A B ⊂ × such that ( , ) a b R ∈ and ( , ) a c R ∈ only if b = c is called a function . In other words, if R is a function, and a dom R ∈ , there is exactly one ordered pair ( , ) a b R ∈ . The second “coordinate” b is thus uniquely determined by a . It is usually denoted R a ( ) . If R A B ⊂ × is a relation, the inverse of R is the relation R B A- ⊂ × 1 defined by R b a a b R- = ∈ 1 {( , ):( , ) } . Example Let A be the set of all people who have ever lived and let S A A ⊂ × be the relation defined by S a b b a = {( , ): } is the mother of . The S is a relation on A , and is, in fact, a function. The relation S- 1 is not a function, and domS A- ≠ 1 . The fact that f A B ⊂ × is a function with dom f = A is frequently indicated by writing f A B : → , and we say f is a function from A to B . Very often a function f is defined by specifying the domain, and giving a recipe for finding f(a) . Thus we may define the function f from the interval [0,1] to the real numbers by f x x ( ) = 2 . This says that f is the collection of all ordered pairs ( , ) x x 2 in which x ∈ [ , ] 01 . Exercises 3.2 1 . Let A be the set of all Georgia Tech students, and let B be the set of real numbers. Define the relation W A B ⊂ × by W a b b a = {( , ): } is the weight (in pounds) of . Is W a function? Is W- 1 a function? Explain. 2 . Let X be set of all states of the U. S., and let Y be the set of all U. S. municipalities. Define the relation c X Y ⊂ × by c x y y x = {( , ): } is the capital of . Explain why c is a function, and find c (Nevada), c (Missouri), and c (Kentucky). 3 . With X , Y as in Exercise 2 , let b be the function b x y y x = {( , ): } is the largest city in . a)What is b (South Carolina)? b)What is b (California)? c)Let f c b = ∩ , where c is the function defined in Exercise 2 . Find dom f ....
View Full Document

This note was uploaded on 10/27/2009 for the course MBA 1918272 taught by Professor Peter during the Fall '09 term at Aberystwyth University.

Page1 / 8

cal3 - 3.1 Chapter Three Vector Functions 3.1 Relations and...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online