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I am not sure how to go about solving this problem. Can somebody tell me how to define $F : Z \to Z$ by the rule $F(n) = 2 -3n$, for all integers $n$ ?

I am not sure where to even start or what is meant by the question. The assignment continues:

i) Is $F$ one-to-one? (Prove or give a counterexample.)

ii) Is $F$ onto? (Prove or give a counterexample.)

Any help that points me in the right direction to go about solving such a problem is greatly appreciated.

cgo
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User
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    ""Define $F$ by..." is not an instruction for you to do anything, it is just telling you what $F$ is. You then have to answer (i) and (ii). – David Apr 24 '15 at 03:39
  • Ah ok, makes sense. Do you know what "one-to-one" and "onto" means? I am assuming one-to-one means that the value n remains the same once it passes through the function? I am not sure what onto means. – User Apr 24 '15 at 03:41
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    I am $99.94$% certain that these terms will have been defined in your lectures/course/book/whatever. Please look them up. If not, they are certainly known to Google. – David Apr 24 '15 at 03:44

1 Answers1

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i) $F$ is one to one. We want to show from $F(a)=F(b)$ we can conclude $a=b$. I will let you see if you can conclude this part. ii) F is not surjective. Counterexample: 0 is an integer but $2-3n \neq 0$ for any integer $n$. Summary: Basically for i) for one-to-one: we want to show there is no element in the codomain that gets hit more than once. ii) for onto: for this we want to show every element of the codomain gets hit at least once

randomgirl
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  • Thank you, this clarifies a lot! How can I tell by the equation F(n) = 2-3n whether or not an element is hit? – User Apr 24 '15 at 04:09
  • Well f(x)=2-3x with codomain and domain all real numbers is just a line. You can look at a visual representation of your graph by erasing a lot of this line. – randomgirl Apr 24 '15 at 04:17
  • Say. $m$ and $n$ are integers and suppose $2-3n=m$ Then $n=\frac{2-m}{3}$. This is a contradiction because $n$ is rational here. Or I should say I guess $n=\frac{2-m}{3}$ isn't an integer for all integer $m$. – randomgirl Apr 24 '15 at 04:30
  • As we see m=0 doesn't get hit because n is 2/3 then... Or m=1 doesn't get hit because n=1/3 then. m=2 does get hit because n=0 then. – randomgirl Apr 24 '15 at 04:31