option b (1,2)

after removing it you get the relation R={(1,1),(2,2),(3,3)}

which is equivalent because it is relative, symmetric, and transitive.

it is relative because for every x belongs to A, (x, x) is in relation R.

it is symmetric because for every (x, y) in R (y, x) is also in R.

it is transitive because for every (x, y) and(y, z) in R (x, z) is in R. (there are no x->y and y->z,so there is no need for x->z)

Verified answer

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To make a relation R an equivalence relation, it must satisfy three properties:

1. Reflexivity: For every element 'a' in A, (a, a) must belong to R.

2. Symmetry: If (a, b) belongs to R, then (b, a) must also belong to R.

3. Transitivity: If (a, b) and (b, c) both belong to R, then (a, c) must also belong to R.

Now, let's evaluate the given relation R = {(1, 1), (1, 2), (2, 2), (3, 3)} against these properties:

1. Reflexivity:

  - (1, 1) is already in R.

  - (2, 2) is already in R.

  - (3, 3) is already in R.

Reflexivity is satisfied for all elements in A, so we don't need to remove any pairs for this property.

2. Symmetry:

  - (1, 2) is in R, but (2, 1) is not in R.

To make R symmetric, we should remove (1, 2) or add (2, 1). In this case, we'll remove (1, 2) to make it symmetric.

3. Transitivity:

  - There are no pairs that violate transitivity, so this property is already satisfied.

After removing (1, 2), the relation R becomes: R = {(1, 1), (2, 2), (3, 3)}.

So, to make R an equivalence relation in A, the pair that should be removed is:

b) (1, 2)