Modern Algebra Problem: Equivalence Relations and Classes

[OhyesOhno]

Homework Statement

There's this one exam problem that I cannot solve... Here it goes:

Consider the set Z x Z⁺. Let R be the relation defined by the following:

for (a,b) and (c,d) in ZxZ⁺, (a,b) R (c,d) if and only if ad = bc, where ab is the product of the two numbers a and b.

a) Prove that R is an equivalence relation Z x Z⁺
b) Show how R partitions Z x Z⁺ and describe the equivalence classes

Homework Equations

For equivalence relations we have to proof that it is reflexive (xRx), symmetric (aRb = bRa) and transitive (aRb bRc hence aRc)

The Attempt at a Solution

I already did part a... I just have trouble on b... how am I supposed to know the equivalence classes of this?

 

[HallsofIvy]

Good! a was the hard part.

Start from the definition of "equivalence class": two elements are in the same class if and only if they are equivalent to each other.

Think about (a, 1). What pairs are equivalent to (a, 1)? that is, what (x,y) satisfy x*1= a*y? (Think about fractions: x/y.)

 

[OhyesOhno]

Any fraction would satisfy (a,1) right? Because if x/y = a/1, then a = x/y so any fraction will do it?