Question

Suppose a is a finite set with n elements. the number of elements and the rank of the largest equivalence relation on a are

Answer 1

Since the options are not given, I am giving an assumed answer.

The correct answer could be - n².

We can use binary relations to solve this.

Equivalence relations include the reflexive, symmetric, and the transitive elements. In the largest equivalence relation, every element will be related to all the other elements.

Suppose n = 2, then A = {1,2}.

So, the largest equivalence relation will be = {(1,1), (1,2), (2,1), (2,2)}

So, the total number of relations is = 4 and 2² = 4.

So, n² will be the rank of the largest equivalence relation.

Answer 2

suppose set  for n=3;
A={1,2,3};
equivalance relation=Reflexive, symmetric, transitive;
largest equivalance relation= {(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2),(3,1),(1,3)};
then number of elements in equivalance relation=9 ;  
that is 3^2=9;
smallest equivalance relation={(1,1),(2,2),(3,3)};
then number of elements in equivalance relation=3;
that is n==3;
so answer is n^2