Proving equivalence relations 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!
 A: As you say, we check whether or not they're reflexive, symmetric, and transitive.


*

*We define $a \sim b$ if $|a|=|b|$ for $a,b \in \mathbb{R}$.  So we check:


*

*Reflexive:  Is $a \sim a$ for all $a \in \mathbb{R}$?  Yes, because $|a|=|a|$.

*Symmetric:  If $a,b \in \mathbb{R}$ and $a \sim b$, does it follow that $b \sim a$?  Yes, because $|a|=|b|$ implies $|b|=|a|$.

*Transitive:  If $a,b,c \in \mathbb{R}$ and $a \sim b$ and $b \sim c$, does it follow that $a \sim c$?  Yes, because $|a|=|b|$ and $|b|=|c|$ implies $|a|=|c|$.


Hence this is an equivalence relation.

*We define $a \sim b$ if $a \leq b$ for $a,b \in \mathbb{R}$.  So we check:


*

*Reflexive:  Is $a \sim a$ for all $a \in \mathbb{R}$?  Yes, because $a \leq a$.

*Symmetric:  If $a,b \in \mathbb{R}$ and $a \sim b$, does it follow that $b \sim a$?  Not in general, because e.g. $2 \leq 3$ but $3 \not\leq 2$.

*Transitive:  If $a,b,c \in \mathbb{R}$ and $a \sim b$ and $b \sim c$, does it follow that $a \sim c$?  Yes, because $a \leq b$ and $b \leq c$ implies $a \leq c$.


We conclude that $\leq$ is not an equivalence relation (since it's not symmetric).
And so on.
A: Reflexive:- (a,a)belongs to 'R' all 'a' belongs to'A'
Symmetric:- (a,b) belongs to 'R' for all 'a' belongs to 'A'
Transitive:- if (a,b) belongs to 'R'
                        (b,c) belongs to 'R'
                  then (c,a) belongs to 'R' 
                for all (a,b,c) belongs to 'A'
Antisymmetric :- 'a' relation 'b' and 'b' relation 'a' IF a=b for all (a,b) belongs total 'A'
