Discrete Mathematics | Set Theory | Relation | Equivalance Relation

Question

which if the following statement is True for every set?

a. $\exists$ a equivalence class that is also a partition set.

b. Every equivalence relation on a set defines a partition of that set.

c. $\exists$ a partition of a set that is also equal to equivalence class of the set on some equivalence relation.

Answer 1

a. Equivalence class is set. A set of equivalence classes is a partition of the base set. So, equivalence class cannot be a partition set.

So, a is false.

b. An equivalence relation has a tendency to partition the base set in to disjoint sets called equivalence classes. Set of these disjoint sets is a partition of base set of that equivalence relation.

So, b is true.

c. Partition of a set is a set of disjoint sets which have all the elements of base set. Thus partition cannot be equal to equivalence class.

So, c is false.

Comments

Hey, thankyou!.

i have doubt on statmenet C
This statement may be TRUE because, if there is a equivalance relation on some set then it should also be a partition set. and statment state's that "there is some equivlance relation on the set."

let A be a set and E1, E2, E3,..... are equivalance classes of set A then

A = {E1xE1}U{E2xE2}U{E3xE3}U ......

Am i right?