- #1
knowLittle
- 312
- 3
Homework Statement
Let ## H = \{ 2^{m} : m \in Z\}##
A relation R defined in ##Q^{+} ## by ##aRb ##, if ## \frac{a}{b} \in H##
a.) Show that R is an equivalence Relation
b.) Describe the elements in the equivalence class [3].
The Attempt at a Solution
For part a, I think I am able to solve it, tell me what you think of my solution:
Assume:
##2^{m} R 2^{m} , m \in Z##
Since, ## 2^{0} \equiv 1 \equiv \frac{2^{m} }{2^{m} } \in H##
So, R is reflexive, for any m in Z.
Assume:
##2^{m} R 2^{n} ##, then ## \frac{ 2^{m}} { 2^{m} }, m , n \in Z##
If ##m<0 \frac{1}{2^{m} 2^{n}} ## , then it still satisfies ##2^{0} , 2^{m+n} \in H##, since m+n is in Z as well.
Without loss of generality for n<0 or both m and n <0.
Thus, ##2^{n} R 2^{m} ## and R is symmetric.
Assume:
##2^{m} R 2^{n} ## and ##2^{n} R 2^{d} ##, since n, m, and d are in Z , they are transitive.
I have problems in part b.
What would an equivalence class [3] mean in this powers of 2 relation?