Which definitions for relation is equivalence relation?

Question

Which of the following definitions for the relation R (defined below) on A would constitute an equivalence relation?
(n.b.. Each tuple (a, 6) in a given relation indicates that a is related to b, and A = {0,1,2 ,3,4))
Select one:

1. R = 4 (0, 0), (0, 1), (0, 3), (1, 0), (1, 2), (2, 0), (2,1), (2, 2), (2,3), (2, 4), (3,1), (3,3), (4, 0), (4,3) }
2. none of these options
3. R = { (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (1,3), (2,1), (3, 2), (4, 0), (4,3) )
4. R = { (0, 1), (0, 4), (1, 0), (1, 1), (1, 4), (2, 0), (3, 4), (4,3), (4,4) }
5. R = 1 (0, 0), (0, 2), (0, 4), (1, 1), (1, 2), (2, 0), (2,1), (2, 4), (3, 0), (3,1), (3, 4), (4,1), (4, 2), (4,3), (4, 4) }
6. R = { (0, 0), (1, 1), (1, 4), (3, 2), (4, 4) }
7. R = { (0, 0), (1, 1), (1, 3), (2, 2), (3,1), (3,3), (4,4) }

Explanation

There is a relation between R and A. And that is in a equivalence type. So it can be as follows:

Relation Reflexive,  This relation means that for all element a∈A, (a,a) ∈ R

Relation Symmetric,  This relation means if all (a,b) ∈ R, (b,a) ∈ R.

Relation Transitive,  This relation means if (a,b) ∈ R and (b,c) ∈ R then (a,c) ∈ R.

So by the information in the above the correct answer for the question is
R={(0,0),(1,1),(1,3),(2,2),(3,1),(3,3),(4,4)}

i.e. the last option.

Proof:

A = {0,1,2,3,4}

As we can see all the relations that are given are available in set R. So we read the definitions above. We know that reflexive relation is relatable here. Also R is the reflexive relation here.

Also we read  symmetric relation. So that we can say that all (a,b) ∈ R, (b,a) ∈ R.   Such a that R is also a Symmetric relation.

                  Next thing is that we also learn the definition of transitive relation. So in this there is a (a,b) ∈ R and (b,c) ∈ R then (a,c) ∈ R. This is in a such a way yhat we can say R is in a transitive relation also. 

So If R is reflexive, symmetric, and transitive relation Hence, it is proved that R is an equivalence relation.

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