Consider the following first order logic formula in which R is a binary relation symbol.
$\forall x \forall y (R (x, y) \Rightarrow R (y, x))$
The formula is

Satisfiable and valid

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Unsatisfiable but its negation is valid

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Satisfiable but its negation is unsatisfiable

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Satisfiable and also its negation

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Solution

The correct option is D Satisfiable and also its negation
Since a relation may or may not be symmetric, the given predicate is satisfiable but not valid .
So (a) is clearly false.
Whennever a predicate is satisfiable its negation also is satisfiable . So option (b) is the correct answer.

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Similar questions

Consider the first-order logic sentence $F : \forall x (\exists y R (x, y))$ . Assuming non-empty logical domains, which of the sentences below are impied by F ?
I $\exists y (\exists x R (x, y))$
II $\exists y (\forall x R (x, y))$
III $\forall y (\exists x R (x, y))$
IV $⇁ \exists x (\forall y ⇁ R (x, y))$

Q. Which of the following first order formulae is logically valid? Here

α (x)

is a first order formula with x as a free variable, and

β

is a first order formula with no free variable.

Q. Consider the binary relation: R = {(x, y), (x, z), (z, x), (z, y)} on the set {x, y, z}. Which one of the following is TRUE?

Q. Which one of these first -order logic formulae is valid?

Q. Which of the following is a valid first order formula? (Here

α

and

β

are first order formulae with x as their only free variable)

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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

The correct option is D Satisfiable and also its negationSince a relation may or may not be symmetric, the given predicate is satisfiable but not valid . So (a) is clearly false. Whennever a predicate is satisfiable its negation also is satisfiable . So option (b) is the correct answer.

Consider the following first order logic formula in which R is a binary relation symbol.
$\forall x \forall y (R (x, y) \Rightarrow R (y, x))$
The formula is

The correct option is D Satisfiable and also its negation
Since a relation may or may not be symmetric, the given predicate is satisfiable but not valid .
So (a) is clearly false.
Whennever a predicate is satisfiable its negation also is satisfiable . So option (b) is the correct answer.