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Is there a way to make a one to one correspondence between the set of even natural numbers and the set of integers not including 0? What would that function be?

Gerry Myerson
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    Welcome to MSE! You should also tell what are your ideas in solving this question. This would not attract downvotes and will prevent the question from being closed due to low quality. For more details on how to write a good question, you may refer here: https://math.meta.stackexchange.com/questions/9959/how-to-ask-a-good-question – Aniruddha Deshmukh Oct 21 '19 at 03:53
  • They are both countable infinite sets, so of course there is a one-to-one correspondence. – Gerry Myerson Oct 21 '19 at 03:54
  • What would it look like? As a function –  Oct 21 '19 at 03:55
  • I can't figure out the piecewise function? –  Oct 21 '19 at 03:56
  • Find a correspondence between the first set and the naturals, and a correspondence between the second set and the naturals, and then do a composition. – Gerry Myerson Oct 21 '19 at 11:47
  • You've had a couple of answers, Sanchit. If you're happy with one of them, you can "accept" it by clicking in the check mark next to it. If you're not happy, you could leave comments to show what you still need. – Gerry Myerson Oct 24 '19 at 11:39
  • You could do that today, Sanchit. – Gerry Myerson Oct 25 '19 at 11:57

2 Answers2

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Take the multiples of $4$ to the positive integers and the numbers equivalent to $2 \bmod 4$ to the negative integers.

Ross Millikan
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  • So any number that is not a multiple of 4 and is an even natural number will have a modulus = 2 when I take x mod 4. So could you express this as a piecewise function so I can see it clearly –  Oct 21 '19 at 04:15
  • Wold it just be: F(x) = {x, if x=4n {-x, if n%4=2 –  Oct 21 '19 at 04:18
  • No, try a few small values and you will find you miss a bunch of integers. $f(2)=-2, f(4)=4,f(6)=-6,f(8)=8, \ldots $ You need a divide by $4$ because the steps in the domain are $4$ and the step you want in the range are $1$. You are getting there. – Ross Millikan Oct 21 '19 at 04:57
  • Oh okay. I see. But how do I fix the negative integers? If I divide by 4 for the x modulus 4 statement, I will get rational numbers. –  Oct 21 '19 at 05:24
  • You add $2$ before dividing by $4$ – Ross Millikan Oct 21 '19 at 14:36
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$$\matrix{x&2&4&6&8&10&\cdots\cr n&1&2&3&4&5&\cdots\cr y&1&-1&2&-2&3&\cdots\cr}$$ If $n$ is odd, then $y=(x+2)/4$; if $n$ is even, then $y=-x/4$.

Gerry Myerson
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