Exhibit a bijection between N and the set of all odd integers greater than 13

  • Thread starter phillyolly
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  • #1
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Homework Statement



Exhibit a bijection between N and the set of all odd integers greater than 13

Homework Equations





The Attempt at a Solution


I didn't have a template for the problem solving. Please check if I did it in the right way? (The way and order a professor will like to see.)
 

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Answers and Replies

  • #2
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Almost...your injection proof is fine. But, when proving a mapping is surjective, we need to show that the domain maps all of the range, i.e., show that for each y, there's an x such that f(x) = y.

You wrote "for y E N," but that's not true. y E B = set of all odd integers greater than 13, not all natural numbers.

So, basically, what you have to do in the surjective proof here is show that if y is an odd integers greater than 13, then x must be a natural number and thus exist in our domain.
 
  • #3
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Should I do this in math induction?
 
  • #4
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hmm, im not sure if i'm right. but when you define a function [tex]f(x)=2x+13[/tex] don't say [tex]for \ all \ x \in N [/tex] yet, because that is what you suppose to show.

so just define this function [tex]f(x)=2x+13[/tex]

like Raskolnikov suggested. any y in the codomain has the form of 2q+13, where q are natural number

so, like you did, we solve for [tex]y=2x+13 \Rightarrow 2q+13=2x+13[/tex], so you want to show that x is natural number ie: x is the set of the domain
 

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