Question

Prove the following is symmetric. Let a ~ b on "all rational numbers" iff a-b=m for some integer m.

Answer 1

I know it is set up as If a~b, then b~ a. But I am stuck on setting up the proof.

Answer 2

Hmm maybe I am misunderstanding, since it seems like this should be antisymmetric because:

a~b iff a-b=m but then we can factor out a negative sign to get: (a-b)=-(b-a) so we have

b-a=-m

so it seems like

a~b = -(b~a)

Answer 3

I fairly sure it is symmetric because i cannot determine any counter examples

Answer 4

I just don't remember how to set up proofs to determine symmetry.

Answer 5

You have misunderstood. We do not require that b-a = m = a - b for b~a when a~b since we only require that b-a=n, for some integer n. Your confusion is that you think that n has to equal to m.

Answer 6

To start your proof:
Let a~b. Then there exists an integer m such that a-b=m. (Now you have to prove that b-a is an integer and so therefore b~a).

Answer 7

OK...

Answer 8

Sorry totally lost on the process. I am OK on reflexive and transitive, but writing proofs for symmetry has me stumped.

Answer 9

Since a and b are integers then b-a is also an integer.

Answer 10

Therefore b ~ a

Answer 11

Would this be correct?

Answer 12

No, since a and b are not necessarily integers, as they could be rational numbers

Answer 13

Open eyes and now I see...the problem is it is Not symmetric and i can find a counter example. thanks for your help.

Answer 14

Nono, a~b DOES imply b~a :P

Answer 15

So, premise is that: a~b
From a~b you must prove that b~a. But to prove that b~a you must prove that b-a is an integer... so basically you have to prove that if a~b, then b-a is an integer.... does that make sense? :P

Answer 16

Yep..let's try this Let b-a = n for some integer n. Then b= n+a then a-(n+a) =m. Then a-n-a =m then -n = m, thus n is an integer,

Answer 17

or should I add |-n| = m, then n is an integer.

Answer 18

Yes, that is basically right! But you could rephrase/organise your proof a bit better... you used an assumption that n is an integer

Answer 19

So if you start from the premise: a~b, then that implies that a-b=m, and therefore b-a=??? :P

Answer 20

b-a = -m

Answer 21

Since - m is an integer then b~a.

Answer 22

Think I make things harder than they need to be.

Answer 23

It's just knowing what the premises are and what conclusion you need to reach. But you got there ^_^

Answer 24

Thank you for your help.

Answer 25

Np :)