Question

Let P(x) be the statement “x can speak Russian” and let $\mathrm{Q}\left(\mathrm{x}\right)$be the statement “ x knowns the computer language C++.” Express each of these sentence in terms of P(x),Q(x), quantifiers, and logical connectives. The domain for quantifiers consists of all students at your school.

a) There is a student at your school who can speak Russian and who knows C++.

b) There is a student at your school who can speak Russian but who doesn’t know C++.

c) Every student at yours school either can speak Russian or knows C++.

d) No student at your school can speak Russian or knows C++.

Short answer

a) There is a student at your school who can speak Russian and who knows C++.

The Sentence can be expressed as$\exists \mathrm{x}\left(\mathrm{P}\left(\mathrm{x}\right)\neg \mathrm{Q}\left(\mathrm{x}\right)\right)$

Step-by-step solution

To Consists of all students in the class

The Given that P(x) is "x can speak Russian" ,is "x Knows the computer language C++,"and 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 is the Statement"x can speak Russian,"Q(x) is the Statement"x knows the computer language C++”also, the Statement" There is a student at your school who can speak Russian but who doesn’t knows C++”

This statement means that there exits a student who speaks Russian and knows C++ this is a case of existential quantifier.

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

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

b) There is a student at your school who can speak Russian but who doesn’t know C++.

The Sentence can be expressed as$\exists \mathrm{x}\left(\mathrm{P}\left(\mathrm{x}\right)\wedge \mathrm{Q}\left(\mathrm{x}\right)\right)$

To Consists of all students in the class

The Given that P(x) is "x can speak Russian" ,Q(x) is "x Knows the computer language C++,"and 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

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

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

Given that P(x) is the Statement"x can speak Russian,"Q(x) is the Statement"x knows the computer language C++”also, the Statement" There is a student at your school who can speak Russian and who knows C++”

This statement means that there exits a student who speaks Russian and knows C++ this is a case of existential quantifier.

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

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

c) Every student at yours school either can speak Russian or knows C++.

The Sentence can be expressed as $\forall \mathrm{x}\left(\mathrm{P}\left(\mathrm{x}\right)\vee \mathrm{Q}\left(\mathrm{x}\right)\right)$

To Consists of all students in the class

The Given that P(x) is "x can speak Russian" , Q(x) is "x Knows the computer language C++," and 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: $\forall xP\left(\mathrm{x}\right)$–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 is the Statement"x can speak Russian,"is the Statement"x knows the computer language C++”also, the Statement" Every student at your school either can speak Russian or knows C++”

This statement means that there exits a student who speaks Russian and knows C++ this is a case of existential quantifier.

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

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

d) No student at your school can speak Russian or knows C++.

The Sentence can be expressed as $\forall \mathrm{x}\left(\mathrm{P}\left(\mathrm{x}\right)\vee \mathrm{Q}\left(\mathrm{x}\right)\right)$

To Consists of all students in the class

The Given that P(x) is "x can speak Russian" , Q(x) is "x Knows the computer language C++," and 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 P(x) is the Statement"x can speak Russian,"Q(x) is the Statement"x knows the computer language C++”also, the Statement"No student at your school either can speak Russian or knows C++”

This statement means that there exits a student who speaks Russian and knows C++ this is a case of existential quantifier.

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

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