Prop: Let ~ be an equivalence relation on a set A, and a, b ∈ A. If a ~ b, then [a] = [b] and if a ≁ b, then [a] ∩ [b] = ∅.
What I understand so far: an equivalence relation has to have 3 properties: reflexive, symmetric, and transitivity. Both a and b are elements of the set A and the questions asking if a and b's cardinality are equal and if a is not an equivalence relation to b, then the intersection of there cardinality must be equal to the empty set.
However I don't quite understand how to write a proof for this proposition, an elementary proof would be absolutely amazing alongside some explanation.
Thank you for all the support math.StackExchange members!
Suppose $[a]\cap[b] \neq \emptyset$. Then there is a $c \in [a]\cap[b]$. Then, by the definition of equivalence classes and intersection, $a\sim c \land c\sim b$. By the transitivity of equivalence relations, $a\sim b$: contradiction.