Empty relation is also valid here because if (x,y) belongs to S become false so the implication become true ie (x,y) belongs to S => x = y become true.

The condition (x,y)∈S⇒x=y implies that every ordered pair in S must be equal. So how come (1,2) (2,3),(3,4) and other such pairs of different numbers can be taken in S? acc to me answer should be 4 only (0,0), (1,1), (2,2), (3,3).

Answer is 2^4=16. The main confusion here is whether to include phi in the relation or not. well, we will count phi.

(x,y)∈S⇒x=y

clearly by law of implication, for some pair say (1,2) , if we try (1,2) ∈S but 1!=2. so this is T->F resulting in F. so (1,2) can not belong to S. now phi is an empty relation it has no element so for phi , (x,y)∈S fails and we know F->anything is True so Phi will be in the relation.