Discrete Math, Difference between $\mathbb Z$ and $\mathbb R$ notation

Question

This may sounds stupid, but in the text book, they ask me a question that is something like this

Define $f:\mathbb{R} \to \mathbb{R}$ by $f(x)= 3x + 2$ and $h: \mathbb{Z} \to \mathbb{Z}$ by $h(n) = 3n+2$.

a) Is f surjective? Prove or give a counter example b) Is h surjective? Prove or give a counter example

I am not sure what is the difference between the notation of these two. I mean I know $\mathbb{R}$ means Real number and $\mathbb{Z}$ means integers. But aren't they just the same answer?

Thanks

score 4 · Answer 1 · answered Feb 05 '14 at 03:33

4

They are not the same answer. Note that $f\left(\dfrac{1}{3}\right)=3$ is allowed while $h\left(\dfrac{1}{3}\right)$ is not allowed as $\dfrac{1}{3}\not\in\mathbb{Z}$. This example answers half of the question for you.

answered Feb 05 '14 at 03:33

John Habert

4,001

Thanks for the common, so would you say a) is surjective and b) is not surjective? – Big Ticket Feb 05 '14 at 03:43
Yes. I've provided the counterexample. You still need to prove $f$ is surjective. – John Habert Feb 05 '14 at 03:44
thank you, I proved that part already but yeah, just not sure about h. – Big Ticket Feb 05 '14 at 03:44

score 0 · Answer 2 · answered Feb 05 '14 at 03:43

When you speak of a function you aren't just speaking of the rule of assignment, you are also speaking of the domain and range. Those are critical in the definition of injectivity and surjectivity.

You are correct in observing that the rules of assignment coincide for elements that lie in both domains, but that doesn't address the question.

Discrete Math, Difference between $\mathbb Z$ and $\mathbb R$ notation

2 Answers2