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

[phillyolly]

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.)

 

[Raskolnikov]

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.

 

[phillyolly]

Should I do this in math induction?

 

[annoymage]

hmm, I am not sure if I'm right. but when you define a function $$f(x)=2x+13$$ don't say $$for \ all \ x \in N$$ yet, because that is what you suppose to show.

so just define this function $$f(x)=2x+13$$

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 $$y=2x+13 \Rightarrow 2q+13=2x+13$$, so you want to show that x is natural number ie: x is the set of the domain