1. ## Relations

Let R be a relation on positive $\displaystyle Z$ defined as follows:

for all a, b as a member of positive $\displaystyle Z$, aRb if and only if a/b = 2 raised to the i for some integer i.

(a) Prove R is a equivalence relation.
(b) Find 3 members of the equivalence class of 60.

To show that R is an equivalence relation, I must show that it's reflexive,symmetric and transitive:

R is reflexive if and only if aRa:

a/a = 2 raised to the i

But we know: a/a = 1, and I don't know how to prove that R is reflexive.

R is symmetric aRb and bRa

a/b = 2 raised to the i and b/a= 2 raised to the i

We can suppose a/b = 2 raised to the i and then try to show that b/a= 2 raised to the i, but thats as far as i've gotten

when R is transitive I'm at a loss...

ALSO, i dont know how to get the equivalency class 60 when its not in the set of numbers 2 raised to the i

help...

2. Originally Posted by luckyNUM7
Let R be a relation on $\displaystyle Z^+$ defined as follows: for all a, b in $\displaystyle Z^+$, aRb if and only if a/b = 2 raised to the i for some integer i.
(a) Prove R is a equivalence relation.
(b) Find 3 members of the equivalence class of 60.
Note that $\displaystyle i$ can be any integer.
So to prove the relation is reflexive let $\displaystyle i=0$ because $\displaystyle 2^0=1$.

Symmetry is as easy: if $\displaystyle \frac{a}{b}=2^j$ then $\displaystyle \frac{b}{a}=2^{-j}$.

You do transitivity.

Because $\displaystyle \frac{60}{15}=2^2$, you now at least one member of $\displaystyle R_{[60]}$.