- #1
mliuzzolino
- 58
- 0
Homework Statement
Consider the relation on [itex]\mathbb{N}[/itex] given by aRb if there exists k [itex] \in [/itex] [itex]\mathbb{Z}[/itex] such that a/b = 2k.
Give an example of two different equivalence classes (that is, find x, y [itex] \in \mathbb{N} [/itex] such that Ex [itex] \neq [/itex] Ey, where Ex and Ey are the equivalence classes of x and y; respectively).
Homework Equations
Ex = {n [itex] \in \mathbb{N} [/itex]: x ~ n}.
Ey = {n [itex] \in \mathbb{N} [/itex]: y ~ n}.
a ~ b by a/b = 2k
The Attempt at a Solution
I'm having an incredibly difficult wrapping my head around the concept of equivalence classes in the context of this problem. I'm not sure where to even begin, so forgive me if I'm off to a very incorrect start...
Let k = 1, so 2 = a/b --> a = 2b where 2 = n [itex] \in \mathbb{N} [/itex]
Then let k = -1, so 1/2 = a/b --> b = 2a where 1/2 = n [itex] \in \mathbb{N} [/itex]
Then a = 2(2a), where a [itex] \neq [/itex] 4a.
Therefore, I have no idea what any of this means...