Find the number of empty relation we can define on a non empty set A

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Solution

This is a question where you get tested for your ability to distinguish number of elements in an empty relation and number of empty relations. Clearly, the number of elements in an empty relation is zero, as no element is connected to any element. But the number of such relations is one, which is denoted by {}. We can define the same empty relation in different ways(using different phrasing and language), but they will all be equal to {}. So the number of empty relation we can define on a set is one.

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How many of the following statements are wrong

1. A relation is said to be reflexive on set A if each element in A is related to itself.

2. Empty relation defined on a non- empty set is reflexive

3. We can define a reflexive relation on A x B, where A is not a subset of B.

4. In a reflexive relation, all the elements should be of the form (a,a) for all a from the set on which the relation is defined

The empty relation defined on a set of real numbers is transitive.

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Empty relation on set A and empty set has the same number of elements

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