Set Theory (III)
Equivalence Classes of an Equivalence Relation
Let ~ be an equivalence relation on a non-empty set A.
Then for any a,b in A
(i) cl(a) is not equal to phi
(ii) Any two equivalence classes cl(a), cl(b) are
either cl(a) intersection cl(b) = phi or, cl(a) = cl(b)
i.e., either cl(a), cl(b) are equal or disjoint.
i.e., two equivalence classes are either equal or have no element in common.
(iii) A = Union cl(a) where a is in A.
The fully formatted problem is in the attached file.
Properties of the equivalence classes of an equivalence relation are defined. The solution is detailed and well presented.