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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


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Solution

We will go through each statement and check if it is correct.

1. This is the definition of reflexive relation. A relation in which each element is related to itself is called a reflexive relation.

2. In a reflexive relation every element will be related to itself. But in an empty relation, no element will be related to any element, which makes it not reflexive.Hence this is false

3. In a reflexive relation, every element should be related to itself. We are given A is a subset of B. This means that not all elements in A will be there in the set B. So, we can say that some elements in A may not be related to the same element in B, because B simply don't have it. This violates the condition for a relation to be symmetric. So the given statement is false

4. Reflexive relation is defined as a relation in which all the elements are related to itself. This does not mean that an element should not be related to any other element. The given statement says all the elements should be of the form (a,a). This means that an element should not be connected to any other element. But in a reflexive function, an element can be related to some other element from the set on which the relation is defined. False

  3 of the given statements are wrong

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