Prove equivalence relation
Please critique the following proof and also let me know if I have defined the equivalence classes correctly.
Let . Define a relation on A by iff . Give a quick proof that this is an equivalence relation. What are the equivalence classes? Explain intuitively.
Proof: Test reflexive for . If then : Given f is a funcion then implies because two different values in the domain cannot be mapped to the same value in the range. Therefore .
Test symmetric: Let where . Then we know . But because = is symmetric we know and since is defined with iff, we know which implies .
Test transitive: Let and where . Then we know and . Because = is transitive we know , but because is defined with iff, we can conclude that . So .QED
The equivalence classes in A would be the sets such that .