Could someone help me with this question? I would like to know if I did it right.
Define the relation ∼ on $\mathbb{R}$ such that x ∼ y if and only if ⌊2x⌋ = ⌊2y⌋.
1) Prove that ∼ is an equivalent relation. Take x,y,z ∈ R. I have the symmetry and reflexivity part, but I´m not sure about the transitivity:
take x ∼ y and y ∼ z, then ⌊2x⌋ = ⌊2y⌋ and ⌊2y⌋ = ⌊2z⌋. It's trivial that ⌊2x⌋ = ⌊2z⌋ and x ∼ z, so the function is transitive?
2) The equivalence classes give a partition of R. Describe the partition.
The partition is ( [i, i + 1/2): i ∈ Z ) of $\mathbb{R}$?
I hope you can help me. Thanks!
Point 1 is ok.
About point 2: I would say that the partition is something like $$ \mathcal{P}=\{A_k: k\in \mathbb{Z}\}, $$ being $A_k=[k,k+\frac{1}{2})$.
Basically is what you have written, but you have to improve the notation. A partition is a set of subsets.