Consider the following first order logic formula in which R is a binary relation symbol.
∀x∀y (R(x, y) => R(y, x))
The formula is
(A) satisfiable and valid
(B) satisfiable and so is its negation
(C) unsatisfiable but its negation is valid
(D) satisfiable but its negation is unsatisfiable
Answer: (B)
Explanation:
VxVy R(x,y) => R(y,x)
The above given relation is symmetry
But, we have both symmetric relations possible and also possibility of anti symmetric relation But neither of always holds for all possibilities of sets.
=> Both are satisfiable but not valid
This solution is contributed by Anil Saikrishna Devarasetty.
One more solution :
We are given a logical formula. So, to be valid it must be a symmetric relation. Hence, Option A is incorrect. Since, it is a logical formula => it is along with it’s negation is satisfiable.
Hence, option B is correct.
This solution is contributed by Mohit Gupta .
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