##### Number of Ordered Pairs

Let S be a set of nelements. The number of ordered pairs in the largest and the smallest equivalence relations on S are:

 A n and n B n2 and n C n2 and 0 D n and 1

A relation is said to be an equivalence relation, when it satisfies 3 properties mainly:

• Reflexive
• Symmetric
• Transitive

Largest equivalence relation on set S is of size n2 , this is the case when relation all the npossible pairs.

Smallest equivalence relation on set S is of size n , this is the case when relation has only reflexive element pairs.

Example: S = (1,3,5)
Largest ordered set are s x s = { (1,1) (1,3) (1,5) (3,1) (3,3) (3,5) (5,1) (5,3) (5,5) } which are 9 which equal to 3^2 = n^2
Smallest ordered set are { (1,1)( 3,3)(5,5)} which are 3 and equals to n. number of elements.

22 Aug 2017 10:20 pm

Nice explanation...thanks

sanjay
4 Sep 2017 12:17 am

What about empty relation ,is it an equivalence relation or not??could anyone explain with an example

deepak
4 Sep 2017 01:02 am

In reflexive relation, all diagonal element have to present And

empty set has no diagonal element;

hence empty relation is not an equivalence relation.