Predicate logic with multiple quantifiers

• MHB
• brynjolf23
In summary: This $y$ does not need to be found beforehand; it can be found by trying all possible $x$ values and seeing which of them makes $P(x,y)$ true.
brynjolf23
Hello everyone. This is my first post on this forum. Thank you for taking the time to help me with my question.
I have no idea where to start. :(

Question 1:
Find an example of a predicate P(x,y) where the domain of x and y are D such that
$\forall x \in D, \exists y \in D, P(x,y)$ is true but $\exists y \in D, \forall x \in D, P(x,y)$ is false.

Let D={1,2,3}

Question 2:
Is it possible to find a predicate P(x,y) such that:
$\exists y \in D, \forall x \in D, P(x,y)$ is true but $\forall x \in D, \exists y \in D, P(x,y)$ is false

Let D={1,2,3}

Question 3:
Is there any method of finding a suitable predicate? or do i just have to guess my way through?

Last edited:
Hi brynjolf23 and welcome to MHB!

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brynjolf23 said:
Hello everyone. This is my first post on this forum. Thank you for taking the time to help me with my question.
I have no idea where to start. :(

Question 1:
Find an example of a predicate P(x,y) where the domain of x and y are D such that
$\forall x \in D, \exists y \in D, P(x,y)$ is true but $\exists y \in D, \forall x \in D, P(x,y)$ is false.

Let D={1,2,3}

Question 2:
Is it possible to find a predicate P(x,y) such that:
$\exists y \in D, \forall x \in D, P(x,y)$ is true but $\forall x \in D, \exists y \in D, P(x,y)$ is false

Let D={1,2,3}

Question 3:
Is there any method of finding a suitable predicate? or do i just have to guess my way through?
Hi,

To be able to help you, we need to know the context of your questions. That is why we ask you to show any work you have done (even if it is wrong); at least, you should tell us what you are studying. This is specially important in formal logic, since there are several methods of deduction.

In this case, to get you started, I can offer an informal description. Both statements are about finding pairs $(x,y)$ that make $P(x,y)$ true. In the first statement, $\forall x \exists y\, P(x,y)$ you can find, for each $x$, at least one $y$ that makes $P(x,y)$ true. Note that each $y$ may depend on $x$.

On the other hand, in the statement $\exists y\forall x\, P(x,y)$, there must be a single $y$ that works for all $x$.

1. What is predicate logic with multiple quantifiers?

Predicate logic with multiple quantifiers is a formal system used in mathematics and computer science to express statements about objects and their properties. It is an extension of basic predicate logic, which uses quantifiers to specify the scope of the variables in the statement.

2. How many types of quantifiers are there in predicate logic with multiple quantifiers?

There are two types of quantifiers in predicate logic with multiple quantifiers: universal quantifiers and existential quantifiers. Universal quantifiers (∀) are used to express that a statement applies to all objects in a given domain, while existential quantifiers (∃) are used to express that a statement applies to at least one object in a given domain.

3. What is the difference between nested quantifiers and multiple quantifiers?

Nested quantifiers are used to express statements with multiple quantifiers where the scope of one quantifier is dependent on the other. On the other hand, multiple quantifiers are used to express statements with multiple quantifiers where the scopes of the quantifiers are independent of each other.

4. How is predicate logic with multiple quantifiers used in mathematics?

Predicate logic with multiple quantifiers is used in mathematics to express mathematical statements in a precise and unambiguous way. It allows mathematicians to reason about objects and their properties, and to prove theorems using logical deduction.

5. What are some common mistakes made when using predicate logic with multiple quantifiers?

Some common mistakes made when using predicate logic with multiple quantifiers include using the wrong type of quantifier, misunderstanding the scope of the quantifiers, and confusing nested and multiple quantifiers. It is important to carefully define the domain and variables in a statement and to use the correct quantifiers to accurately express the intended meaning.

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