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Let X be a finite set. Describe the equivalence relation having the greatest number of distinct...

Let X be a finite set. Describe the equivalence relation having the greatest number of distinct equivalence classes, and the one with the smallest number of equivalence classes.

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Expert Solution

Let be a finite set with elements then

A relation on the set is equivalnce if it satisfies

1) Reflexive Property :

2) Symmetric Property : If then

3) Transitive Property : and then

Each element of the set has an equivalence class

The equivalence class an element is the set of all those elements of the set with which the element has a relation .

The equivalence class of an element is denoted by .

The equivalence relation has the smallest number of equivalence classes .

The smallest equivalence relation must always contain the diagonal that is

Since every element is related to itself therefor it contains equivalence classes.

Since the set is non empty it contains elements. So, the smallest number of equivalence classes is equal to .

The largest number of equivalence classes in the set which contains elements is equal to because the largest relation in a set is the cartition product that is it contains elements.

For example the set with the relation .

The smallest ordered set is which has three elements that is the smallest number of equivalnce classes.

The largest ordered set of relation is

Which has elements which is the largest number of equivalence classes.

milcah answered 1 year ago

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