Question

Express the negations of each of these statements so that all negation symbols immediately precede predicates.

1. \(\forall x\exists y\forall zT\left( {x,y,z} \right)\)
2. \(\forall x\exists yP\left( {x,y} \right) \vee \forall x\exists yQ\left( {x,y} \right)\)
3. \(\forall x\exists y\left( {P\left( {x,y} \right) \wedge \exists zR\left( {x,y,z} \right)} \right)\)
4. \(\forall x\exists y\left( {P\left( {x,y} \right) \to Q\left( {x,y} \right)} \right)\)

Short answer

The Express the negations statement can be Precede predicates.

Step-by-step solution

Finding the Truth values for \(\forall x\exists y\forall zT\left( {x,y,z} \right)\)

The Given Statement “\(\forall x\exists y\forall zT\left( {x,y,z} \right)\)”

On Applying the negation upon the quantifiers,\(\forall \)quantifier changes to\(\exists \)quantifier and vice-versa that is, quantifier\(\exists \)changes to\(\forall \)quantifier.

Consider the statement as\(\forall x\exists y\forall zT\left( {x,y,z} \right)\)

The Negation of the above statement is\(\exists x\forall y\exists z\neg T\left( {x,y,z} \right)\)

As a result, The Negation of the given statement is as follows.

Finding the Truth values for \(\forall x\exists yP\left( {x,y} \right) \vee \forall x\exists yQ\left( {x,y} \right)\)

The Given Statement “\(\forall x\exists yP\left( {x,y} \right) \cup \forall x\exists yQ\left( {x,y} \right)\)”

On Applying the negation upon the quantifiers,\(\forall \)quantifier changes to\(\exists \)quantifier and vice-versa that is, quantifier\(\exists \)changes to\(\forall \)quantifier.

Consider the statement as\(\forall x\exists yP\left( {x,y} \right) \cup \forall x\exists yQ\left( {x,y} \right)\)

The Negation of the above statement is\(\exists x\forall y\neg P\left( {x,y} \right) \cap \exists x\forall y\neg Q\left( {x,y} \right)\)

As a result, The Negation of the given statement is as follows.

\(\exists x\forall y\neg P\left( {x,y} \right) \cap \exists x\forall y\neg Q\left( {x,y} \right)\)

Finding the Truth values for \(\forall x\exists y\left( {P\left( {x,y} \right) \wedge \exists zR\left( {x,y,z} \right)} \right)\)

The Given Statement “\(\forall x\exists y\left( {P\left( {x,y} \right) \cap \exists xR\left( {x,y,z} \right)} \right)\)”

On Applying the negation upon the quantifiers,\(\forall \)quantifier changes to\(\exists \)quantifier and vice-versa that is, quantifier\(\exists \)changes to\(\forall \)quantifier.

Consider the statement as\(\forall x\exists y\left( {P\left( {x,y} \right) \cap \exists xR\left( {x,y,z} \right)} \right)\)

The Negation of the above statement is\(\exists x\forall y\left( {\neg P\left( {x,y} \right) \cap \exists z\neg R\left( {x,y,z} \right)} \right)\)

As a result, The Negation of the given statement is as follows.

\(\exists x\forall y\left( {\neg P\left( {x,y} \right) \cap \exists z\neg R\left( {x,y,z} \right)} \right)\)

Finding the Truth values for \(\forall x\exists y\left( {P\left( {x,y} \right) \to Q\left( {x,y} \right)} \right)\)

The Given Statement “\(\forall x\exists y\left( {P\left( {x,y} \right) \to Q\left( {x,y} \right)} \right)\)”

On Applying the negation upon the quantifiers,\(\forall \)quantifier changes to\(\exists \)quantifier and vice-versa that is, quantifier\(\exists \)changes to\(\forall \)quantifier.

Consider the statement as\(\forall x\exists y\left( {P\left( {x,y} \right) \to Q\left( {x,y} \right)} \right)\)

The Negation of the above statement is\(\exists x\forall y\left( {P\left( {x,y} \right) \cap \neg Q\left( {x,y} \right)} \right)\)

As a result, The Negation of the given statement is as follows.

\(\exists x\forall y\left( {P\left( {x,y} \right) \cap \neg Q\left( {x,y} \right)} \right)\)