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 relastions possible and also possibility of anti symmetric relation But neither of always holds for all possibilites 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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