# GATE | GATE IT 2006 | Question 21

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

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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