Equivalence Relations in Set Theory: Homework Statement and Solutions

[imranq]

Homework Statement

Homework Equations

An equivalence relation on a set A, is for a,b,c in A if:
a~a
a~b => b~a
a~b and b~c => a~c

The Attempt at a Solution

It seems uncomplicated, but I don't know how I would write down a proof. The book I'm using is Topics in Algebra, 1st Edition Herstein

 

[jgens]

What if there were some element $a \in A$ which wasn't related to any other member of $A$?

 

[HallsofIvy]

In other words, consider A= {1, 2, 3} and the relation is {(1, 1), (1,2), (2,1), (2,2)},

 

[imranq]

In other words, consider A= {1, 2, 3} and the relation is {(1, 1), (1,2), (2,1), (2,2)},

I don't understand this. I think equivalence classes are generalized equal signs for some property. So (1,1) I understand, but how so for (1,2)?