1. ## Prove

Define a relation ~ on N by a~b iff a=b*5^k for some k that is a member of Z.

a) Prove that ~ is an equivalence relation on set N
b)Give a complete set of equivalence class representatives.

2. Originally Posted by mandy123
Define a relation ~ on N by a~b iff a=b*5^k for some k that is a member of Z.
a) Prove that ~ is an equivalence relation on set N
b)Give a complete set of equivalence class representatives.
$\displaystyle \begin{array}{l} a = a5^0 \\ a = b5^k \Rightarrow \quad b = a5^{ - k} \\ a = b5^k \,\& \,b = c5^j \Rightarrow \quad a = c^{k + j} \\ \end{array}$

Are 4, 23, 65 all in the same equivalence class?
Are 5, 50, 75 all in the same equivalence class?

3. ok so how in the world did you figure out the first part?? I am so confused with how you went from the variables a,b,k to the variables c and j?

The second part (b) will those be the equivalence class representatives.
I know that 4,23,65 are not the same equiv. classes, but 5, 50, 75 are the same equiv classes. Right?

4. Originally Posted by mandy123
ok so how in the world did you figure out the first part?? I am so confused with how you went from the variables a,b,k to the variables c and j?
Do you know what equivalence relation is?
If you do, then you recognize that as a proof that the relation is an equivalence relation.
By the way, a, b & c are natural numbers. While k & j are of course integers.

Originally Posted by mandy123
The second part (b) will those be the equivalence class representatives.
I know that 4,23,65 are not the same equiv. classes, but 5, 50, 75 are the same equiv classes. Right?
Do you really know that?
Name us one other number in the equivalence class determined by 4.
How many equivalence classes do you think there are?