Can someone double-check this simple binary relation proof?

[Syrus]

Homework Statement

*attached

Homework Equations

The Attempt at a Solution

Let a,b ∈ H. Then (∀x ∈ S)(a*x = x*a) and (∀x ∈ S)(b*x = x*b). It is easy to see, then, that a*b = b*a. Now let c ∈ S. Then (a*b)*c = c*(a*b) by the associativity of *.

Q.E.D.

 

[micromass]

Homework Statement

*attached

Homework Equations

The Attempt at a Solution

Let a,b ∈ H. Then (∀x ∈ S)(a*x = x*a) and (∀x ∈ S)(b*x = x*b). It is easy to see, then, that a*b = b*a. Now let c ∈ S. Then (a*b)*c = c*(a*b) by the associativity of *.

Q.E.D.

The last statement does not follow from associativity. Associativity merely says that

$$(a*b)*c=a*(b*c)$$

I also see no reason to take c in S. You need to prove that a*b in S. That is, you need to show for all c that (a*b)*c=c*(a*b). You don't only need to prove it for the c in S.

 

[Syrus]

Hey micromass. In order to show that a*b is an element of H, isn't the statement to prove: (for all c in S)((a*b)*c = c*(a*b))? Am i mistaken here?

*By the way i am taking c in S to be an arbitrary element

 

[micromass]

Hey micromass. In order to show that a*b is an element of H, isn't the statement to prove: (for all c in S)((a*b)*c = c*(a*b))? Am i mistaken here?

*By the way i am taking c in S to be an arbitrary element

Oops, I had my notation mixed up Yes, that is correct!

 

[Syrus]

Heh. So wait... is the proof correct?

 

[micromass]

Heh. So wait... is the proof correct?

You still need to explain why (a*b)*c = c*(a*b). It doesn't follow from associativity.

 

[Syrus]

Can i say that since a,b are elements of S (since H is a subset of S), then a*b is in S. But also, since c is in S, then (a*b)*c is in S. and since * is associative on S, a*(b*c) is in S?

 

[micromass]

How does thart prove (a*b)*c=c*(a*b)??

You need to use that a and b are in H.

 

[Syrus]

ahhh, finally Figured it out micromass! I wasn't using the fact that a and b were in H to commute the result. That is, since (a*b)*c = a*(b*c), for some unknown reason I thought this was the goal. But now we may use that a is in H to write (b*c)*a and continue to use the fact that a,b are in H along with associativity to obtain our result.

I appreciate your help