Let A abcde and let C abc de For xy A x y if and only if th

Let A= {a,b,c,d,e} and let C= {{a,b,c}, {d,e}}. For x,y A, x ~ y if and only if there exists a set U in C such that x U and y C. Let a A and U C such that a U. Prove that [a]= U.

Solution

Note that A is a set of elements where as C is a set of sets.

And an equivalence relation for x,y A, i.e. x ~ y exists if and only if there exists a set U in C such that x U and y C.

Now since U C => y U so that x,y U hence any such equivalence class for an element ‘a’ A is the set U, i.e. [a] =U … (Hence proved)

Hope this helps!!