This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: n + 1 whenever it holds for n , the result is true for all n ∈ N + , by induction. 1 3. (10 points) Prove that the function f : R + → R + given by f ( x ) = 1 x is onetoone and onto. Proof. First, we will prove that f is onetoone. Let x 1 , x 2 ∈ R + , and assume that f ( x 1 ) = f ( x 2 ). Then 1 x 1 = 1 x 2 , and therefore, by crossmultiplying, we conclude that x 1 = x 2 . This shows that f is onetoone. Now, we will prove that f is onto. Let y ∈ R + . Then deﬁne x = 1 /y . Since y ∈ R + , y 6 = 0, so x ∈ R , and since y > 0, x > 0. Therefore, x ∈ R + . Finally, f ( x ) = f ± 1 y ! = 1 ± 1 y ! = y, as required. This shows that f : R + → R + is onto....
View
Full
Document
This note was uploaded on 04/04/2010 for the course MAT 300 taught by Professor Thieme during the Spring '07 term at ASU.
 Spring '07
 thieme
 Math, Sets

Click to edit the document details