Ok, here's my problem:

Let X={1,2,...,10}. Define a relation R on X*X by (a,b)R(c,d) if ad=bc.

A. Prove R an equivalence relation.

B. List the elements of [(5,4)].

C. How many distinct equivalence classes are there?

Ok, so here's what I've got so far:

Part A, Prove R is an equivalence relation. So I must prove that R is reflexive, symmetric and transitive.

Reflexive:

I need to show that ((a,b),(a,b)) is an element of R, for the set X. So I will check that ab=ba. Since c=a and d=b, then ab=ba, thus R is reflexive.

Symmetric:

I must show that (a,b),(c,d) is an element of R and (c,d),(a,b) is contained in R. If (a,b),(c,d) is an element of R, then ad=bc and cb=da since multiplication is commutative. Therefore (c,d),(a,b) is contained in R.

Transitive:

I must show that if (a,b),(c,d) is contained in R and (c,d),(e,f) is contained in R then (a,b),(e,f) is contained in R. Assume that (a,b),(c,d) is contained in R and (c,d),(e,f) is contained in R. Then ad=cb and cf=ed. This implies that a/b=c/d and that c/d=e/f so a/b=e/f thus, af=eb. So, (a,b),(e,f) is contained in R.

Part B:

List the elements of [(5,4)]:

I'm confused on this a bit, I am assuming they are asking what elements does the equivalence class [(5,4)] contain...

{(5,4),....

My thoughts here, if (a,b)=(5,4) and ad=bc, then 5d=4c, c=4/5d and d=5/4c, allowing that if c=5, d=4 and if c=10 then d=8.

so, {(5,4), (10,8)} would be my answer. Is this correct?

Ok and Part C...

This I'm lost on how to figure it up. So any help or pointers in the correct direction would be great!

Thanks in advance

Ashley