Question

Let C(x) be the statement “x has a cat” let D(x) be the statement “ x has a dog,” and let F(x) be the statement “ x has a ferret,” Express each of these statements in terms of C(x),D(x),F(x), quantifiers, and logical connectives. Let the domain consist of all students in your class.

a) A student in your class has a cat, a dog, and a ferret.

b)All students in your class have a cat, a dog, or a ferret.

c)Some student in your class has a cat and a ferret, but not a dog.

d)No student in your class has a cat, dogs, and ferrets.

e)For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has this animal as a pet.

Short answer

(a) $\exists \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\wedge \mathrm{D}\left(\mathrm{x}\right)\wedge \left(\mathrm{x}\right)\right)$

(b)$\forall \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\vee \mathrm{D}\left(\mathrm{x}\right)\vee \left(\mathrm{x}\right)\right)$

(c)$\exists \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\wedge \mathrm{F}\left(\mathrm{x}\right)\wedge \neg \left(\mathrm{x}\right)\right)$

(d)$\neg \exists \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\wedge \mathrm{F}\left(\mathrm{x}\right)\wedge \neg \mathrm{D}\left(\mathrm{x}\right)\right)$

(e)$\left(\exists \mathrm{xC}\left(\mathrm{x}\right)\right)\wedge \left(\exists x\mathrm{D}\left(\mathrm{x}\right)\right)\wedge \left(\exists x\mathrm{F}\left(\mathrm{x}\right)\right)$

Step-by-step solution

To Consist of all students in the class

The Given that C(x) is "x has a cat" ,D(x) is "x has a dog," and F(x) is "x has a ferret," the domain consists all students in the class.

The Concept that used the value of the propositional function P at x is said to be a statement P(x). Once a value has been assigned to variable x and a truth value has been determined, the statement P(x) becomes propositional.

Apply the two types of quantifiers

a) The Universal Quantifier: $\forall \mathrm{xP}\left(\mathrm{x}\right)$–The statement is true for at least one element in the domain.

b) The Existential Quantifier: $\exists \mathrm{xP}\left(\mathrm{x}\right)$– The statement holds true for all of the domain’s values.

Given that, C(x) is the statement “x has a cat”

D(x) is the statement "x has a dog," F(x) is "x has a ferret,".

Also, the Statement is “A Student in your class has a cat, a dog and a ferret”

This statement means that there exists a student who has all the three animals this is a case of existential quantifier.

The Statement can be expressed as follows:$\exists \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\wedge \mathrm{D}\left(\mathrm{x}\right)\wedge \left(\mathrm{x}\right)\right)$

As a result, The sentence can be expressed as .$\exists \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\wedge \mathrm{D}\left(\mathrm{x}\right)\wedge \left(\mathrm{x}\right)\right)$

Apply the two types of quantifiers

a) The Universal Quantifier:$\forall \mathrm{xP}\left(\mathrm{x}\right)$–The statement is true for at least one element in the domain.

b) The Existential Quantifier:$\exists \mathrm{xP}\left(\mathrm{x}\right)$– The statement holds true for all of the domain’s values.

Given that ,C(x) is the statement “x has a cat”

D(x)is the statement "x has a dog,"F(x) is "x has a ferret,".

Also, the Statement is “All students in your class have a cat, a dog, or a ferret.”

This statement means that there exists a student who has all the three animals this is a case of existential quantifier.

The Statement can be expressed as follows:$\forall \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\vee \mathrm{D}\left(\mathrm{x}\right)\vee \left(\mathrm{x}\right)\right)$

As a result, The sentence can be expressed as .$\forall \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\vee \mathrm{D}\left(\mathrm{x}\right)\vee \left(\mathrm{x}\right)\right)$

Apply the two types of quantifiers

a) The Universal Quantifier: $\forall \mathrm{xP}\left(\mathrm{x}\right)$–The statement is true for at least one element in the domain.

b) The Existential Quantifier:$\exists \mathrm{xP}\left(\mathrm{x}\right)$– The statement holds true for all of the domain’s values.

Given that , C(x) is the statement “x has a cat”

D(x) is the statement "x has a dog," F(x) is "x has a ferret,".

Also, the Statement is “Some student in your class has a cat and a ferret, but not a dog.”

This statement means that there exists a student who has all the three animals this is a case of existential quantifier.

The Statement can be expressed as follows:$\exists \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\wedge \mathrm{F}\left(\mathrm{x}\right)\wedge \neg \left(\mathrm{x}\right)\right)$

As a result,The sentence can be expressed as .$\exists \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\wedge \mathrm{F}\left(\mathrm{x}\right)\wedge \neg \left(\mathrm{x}\right)\right)$

Apply the two types of quantifiers

a) The Universal Quantifier:$\forall \mathrm{xP}\left(\mathrm{x}\right)$–The statement is true for at least one element in the domain.

b) The Existential Quantifier:$\exists \mathrm{xP}\left(\mathrm{x}\right)$– The statement holds true for all of the domain’s values.

Given that , C(x) is the statement “x has a cat”

D(x) is the statement "x has a dog," F(x) is "x has a ferret,".

Also, the Statement is “No student in your class has a cat, dogs, and ferrets.”

This statement means that there exists a student who has all the three animals this is a case of existential quantifier.

The Statement can be expressed as follows:$\neg \exists \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\wedge \mathrm{F}\left(\mathrm{x}\right)\wedge \neg \mathrm{D}\left(\mathrm{x}\right)\right)$

As a result,The sentence can be expressed as .$\neg \exists \mathrm{x}\left(\mathrm{C}\left(\mathrm{x}\right)\wedge \mathrm{F}\left(\mathrm{x}\right)\wedge \neg \mathrm{D}\left(\mathrm{x}\right)\right)$

Apply the two types of quantifiers

a) The Universal Quantifier: $\forall \mathrm{xP}\left(\mathrm{x}\right)$–The statement is true for at least one element in the domain.

b) The Existential Quantifier: $\exists \mathrm{xP}\left(\mathrm{x}\right)$– The statement holds true for all of the domain’s values.

Given that , C(x) is the statement “x has a cat”

D(x) is the statement "x has a dog," F(x) is "x has a ferret,".

Also, the Statement is “For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has this animal as a pet.”

This statement means that there exists a student who has all the three animals this is a case of existential quantifier.

The Statement can be expressed as follows:$\left(\exists xC\left(\mathrm{x}\right)\right)\wedge \left(\exists x\mathrm{D}\left(\mathrm{x}\right)\right)\wedge \left(\exists x\mathrm{F}\left(\mathrm{x}\right)\right)$

As a result, the sentence can be expressed as .$\left(\exists xC\left(\mathrm{x}\right)\right)\wedge \left(\exists x\mathrm{D}\left(\mathrm{x}\right)\right)\wedge \left(\exists x\mathrm{F}\left(\mathrm{x}\right)\right)$