Non-trivial example of Quantifiers

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In summary, the statement \forall x P(x) \rightarrow \exists x P(x) is a true statement, but it can fail if the universe of discourse is empty. This is known as the trivial case. In most cases, this statement would be true as long as there exists at least one x that satisfies P(x).
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Dragonfall
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Can I have a non-trivial example of where [itex]\forall x P(x) \rightarrow \exists x P(x)[/itex] fails?
 
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  • #2
No, because it is a true statement.
Assume that ##\forall x P(x)##. Let x_0 be arbitrary. Then ##P(x_0)##. ##\exists x P(x)##. QED.

The only way this can fail is if the universe of discourse is empty, in which case ##\forall x P(x)## is true and ##\exists x P(x)## is false, but I guess this is what you call the trivial case.
 
  • #3
Dragonfall said:
Can I have a non-trivial example of where [itex]\forall x P(x) \rightarrow \exists x P(x)[/itex] fails?

I don't know much formal logic at all, but under any logic that matches my intuition, [itex]\forall x P(x) \rightarrow \exists x P(x)[/itex] would be true whenever [itex]\exists x[/itex] is true.
 
  • #4
Then I'm very confused. I thought no-NP languages are decision problems of the form [itex]\forall x P(x)[/itex] and NP languages are [itex]\exists x P(x)[/itex].
 
  • #5
Wait, I figured it out. Nevermind.
 

FAQ: Non-trivial example of Quantifiers

What is a non-trivial example of quantifiers?

A non-trivial example of quantifiers is the statement "There exists an x such that x is a prime number." This statement uses the existential quantifier "there exists" to assert the existence of at least one x that satisfies the condition of being a prime number.

How are quantifiers used in mathematics?

Quantifiers are used in mathematics to express generalizations or statements about a group of objects. They allow us to make claims about all or some members of a set, and are essential in defining and proving mathematical theorems.

What is the difference between universal and existential quantifiers?

The universal quantifier, denoted by the symbol ∀, means "for all" and is used to make a statement about every element in a set. The existential quantifier, denoted by the symbol ∃, means "there exists" and is used to make a statement about at least one element in a set.

Can quantifiers be used in everyday language?

Yes, quantifiers can be used in everyday language to make generalizations or statements about groups of objects. For example, the statement "All dogs have four legs" uses the universal quantifier "all" to make a claim about every dog.

How do quantifiers relate to logic?

Quantifiers are an important component of mathematical logic. They allow us to express statements in a more precise and rigorous manner, and are used in the formalization of mathematical proofs. In logic, quantifiers are also used to construct logical formulas and to determine the truth value of statements.

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