I'm studying for an exam, and I have found question that I'm not sure about how to solve it. The question is: Let A be finite set. Let R be equivalence relation on A.
1.Write necessary and sufficient condition on the cardinal number of quotient set so R will be equality relation (equality relation is the relation R: xRy iff x=y.
2.Is the necessary and sufficient condition from (1.) true for infinite set A?
My solution for 1 is:
|A/R| = |A| iff R is the equality relation.
I think its true because the number of all equivalence classes equal to the the number of elements in A.
For example A = {1,2,3}
[1]R = {1}
[2]R = {2}
[3]R = {3}
My solution for 2 is:
Yes, The cardinal of A/R is still equal to the cardinal of A.
I would like to know where my mistakes are. Tell me please if the absence of Latex notations is a problem.
Your solution to 1) is correct. However for an infinite set the condition is necessary but not sufficient. For example, consider the set of integers under an equivalence relation where all the equivalence classes are finite sets. (Thanks to Andreas Blass for correction.)