@Deepak Poonia sir plz correct me ,if I go somewhere wrong.
What is model?
→ Model means an Interpretation which makes your formula True.
Interpretation means (Domain + Predicate).
What is satisfiable?
→ If there exist atleast one model means if there exist atleast one domain for which my statement is true.
What is valid?
→ If we consider every domain(finite domain and infinite domain) and for every domain my statement is true.Then we can say that this is valid
Now, R(x,y) : x divides y.
So,R(y,x): y divides x.
Let consider my domain , D = {2}
∀x∀y(R(x, y) ⟹ R(y, x))
→ R(2,2) will be true as 2 divides 2. So,there exist a model.So, this is satisfiable.
- Let consider my domain , D = {2}
R(2,2) is true and ~R(2,2) will be false.
Now the negation of ( ∀x∀y(R(x, y) ⟹ R(y, x)) ) will be ∃x∃y(R(x,y)Λ~ R(y,x))
And so obvious that the expression will be false for this domain D. So the negation is surely not valid.
- Let consider my domain , D = {2,4}. R(2,4) is true and R(4,2) will be false. So ~R(4,2) will be true. ∃x∃y(R(x,y)Λ~ R(y,x)) this expression will be model.
So the negation will never be unsatisfiable for this expression.
We can conclude answer is option B.