74-hw5s - MATH 74 HOMEWORK 5 SOLUTIONS 1 Just so people can...

Info iconThis preview shows pages 1–2. 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: MATH 74 HOMEWORK 5 SOLUTIONS 1. Just so people can see how these go I am including proofs. They were not a required part of the homework. 1(a). Injective but not surjective. Proof. To see that f is not surjective, notice that 2 is not in the range of f . Indeed, 1 2 = 1 is less than 2, and if x ≥ 2, then x 2 ≥ 4 is bigger than 2, so there is no n ∈ N for which n 2 = 2. To see that f is injective, fix x,y ∈ N and suppose f ( x ) = f ( y ), ie, that x 2 = y 2 . We cannot have x < y because then x 2 = x · x < x · y < y · y = y 2 , and similarly (interchanging x and y in the proof just given) we cannot have x > y . So x = y . There is a general fact at work here. The point is that squaring, on the given domain, is strictly increasing : if x < y , then f ( x ) < f ( y ). Any strictly increasing function is injective. 1(b). Injective but not surjective. Proof. To see that it is injective, consider the function g : R → R given by g ( x ) = x 1 / 3 . Then g ( f ( x )) = ( x 3 ) 1 / 3 = x, x ∈ R . so f is injective by “Theorem 18” from class. To see that f is not surjective, notice that every number in the range of f , being a cube of a nonnegative number, must be nonnegative. So any negative number (eg- 1) is not in the range of f . 1(c). Injective and surjective. Proof. Without a precise definition of the sine function (or the number π ) we are not really in a position to “prove” this, but it is clear from calculus. On the givennot really in a position to “prove” this, but it is clear from calculus....
View Full Document

This homework help was uploaded on 04/02/2008 for the course MATH 74 taught by Professor Courtney during the Fall '07 term at Berkeley.

Page1 / 3

74-hw5s - MATH 74 HOMEWORK 5 SOLUTIONS 1 Just so people can...

This preview shows document pages 1 - 2. Sign up to view the full document.

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