I just started my abstract algebra class and I am struggling with the concept of equivalence relations. I know that in order to prove equivalence relations, I have to prove the reflexive, symmetric, and transitive properties. However, I don't know how to go about starting the actual proof or solution. I have these examples and any help would be appreciated.
I have to show which of the following are equivalence relations on the set of real numbers and, if they are not, why.
- $a\sim b$ iff $|a|=|b|$
- $a\sim b$ iff $a\leq b$
- $a\sim b$ iff $|a-b| \leq 1$
Thank you for any help!