Let S denote all integers, define a relation R on S as aRb if ab ≥ 0 where a, b ∈ S’. Then R is :

Concept:

1. Reflexive: Each element is related to itself.

• R is reflexive if for all x ∈ A, xRx.

2. Symmetric: If any one element is related to any other element, then the second element is related to the first.

• R is symmetric if for all x, y ∈ A, if xRy, then yRx.

3. Transitive: If any one element is related to a second and that second element is related to a third, then the first element is related to the third.

• R is transitive if for all x, y, z ∈ A, if xRy and yRz, then xRz.

4. R is an equivalence relation if A is nonempty and R is reflexive, symmetric and transitive.

Calculation:

S = Set of all integers and R = {(a, b), a, b ϵ  S and ab  0}

For reflexive:

aRa = a.a = a² ≥ 0, so it's reflexive.

For symmetric:

aRb = ab ≥ 0 and bRa = ba ≥  0, So relation is symmetric

For transitive:

For all integers, if ab ≥ 0, bc ≥  0, then for all the cases ac ≥ 0 is not true.

For example,

If a = -1, b = 0, and c =1

Then,  ab ≥ 0, bc ≥  0 but for ac = -1 which is not satisfying ac ≥ 0 so R is not transitive.

So, Relation Is reflexive, symmetric but not transitive. 

Hence, option (2) is correct.