Let F be the following set F = ${(... \pm\frac{1}{2^3},\pm\frac{1}{2^2},\pm\frac{1}{2},0,1,\pm2,\pm2^2,\pm2^3....)}$

Question

Not sure if I formatted it correctly, this is a screenshot of the question https://gyazo.com/75bb2ea6feb5babba9e60e081cdf8fb8

Let F be the following set

$F = {{ (... \pm \frac{1}{2^3},\pm\frac{1}{2^2},\pm\frac{1}{2},0,1,\pm2,\pm2^2,\pm2^3...)}}$

Is this set (with the usual addition and multiplication a field? Explain.

I attempted to disprove this using a single counter-example. I would like to know whether or not this would be considered correct, or if I am even on the right track.

This set is not a field. It does not follow (1), a counter-proof to this would be adding the elements +2 and +2^2

By (1), x+y∈F

(2)+(2^2 )∈F→2+4=6,6∉F

∴ This set is not a field.

Answer 1

Yes, that is correct. – Kevin Long 10 mins ago

I would start by saying, "Assume FF is a field," so that it's clear you're beginning a proof by contradiction, and then continuing with, "By (1)..." but it works! – user474330 9 mins ago

Answer 2

Once you have 0 and 1 you get, by repeated addition, all the integers.

Once you have all the integers, by division, you get all the rationals.

So what you have is definitely not a field.