Define equivalence relation on set of real numbers so distinct equivalence classes are $[2k,2k+2)$ Define an equivalence relation $\sim$ on $\mathbb{R}$ such that the distinct equivalence classes of $\sim$ are $[2k,2k+2)$, where $k$ is an integer (Hint: find an appropriate function $f$ with all real numbers as its domain and let $R = \{(x,y)|f(x)=f(y)\}$.)
I feel like I can do these problems, but starting out I almost never know what is being asked. I am so utterly confused. I know what an equivalence relation is, and I know what equivalence classes are, but how can equivalence classes be $[2k,2k+2)$ with $k $ integer? I think I'm confusing myself further.

RSerrao has already given a good answer to this question. If you are still confused about how intervals can be equivalence classes, remember that intervals are just sets of real numbers; for example, $[1,2)$ is the set $\{x \in \mathbb{R}|1\le x<2\}$. Since equivalence classes are sets, it is fine for intervals to be equivalence classes. I hope this clarifies the second part of your question!
