Question

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

1. There is a student at your school who can speak Russian and who knows C++.
2. There is a student at your school who can speak Russian but who doesn’t know C++.
3. Every student at your school either can speak Russian or knows C++.
4. No student at your school can speak Russian or knows C++.

Short answer

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

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

Step-by-step solution

To Consist of all students in the class

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

1. The Universal Quantifier:$\exists xP\left(x\right)$–The statement is true for at least one element in the domain.
2. The Existential Quantifier:$\exists xP\left(x\right)$– The statement holds true for all of the domain’s values.

Given that P(x) is the Statement "x can speak Russian," 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 know C++”

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

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

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

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

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

To Consist of all students in the class

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

1. The Universal Quantifier:$\forall xP\left(x\right)$–The statement is true for at least one element in the domain.
2. The Existential Quantifier: $\exists xP\left(x\right)$– The statement holds true for all of the domain’s values.

Given that P(x) is the Statement"x can speak Russian," 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 exists a student who speaks Russian and knows C++ this is a case of the existential quantifier.

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

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

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

The Sentence can be expressed as$\forall x\left(P\left(x\right)\neg Q\left(x\right)\right)$

To Consist of all students in the class

Given that p(x) is "x can speak Russian", Q(x) is "x Knows the computer language C++," and the domain consists of 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

1. The Universal Quantifier:$\forall xP\left(x\right)$–The statement is true for at least one element in the domain.
2. The Existential Quantifier:$\exists x\left(P\left(x\right)\vee Q\left(x\right)\right)$– 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 "Every student at your school either can speak Russian or knows C++”

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

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

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

d) No student at your school can speak Russian or knows C++.The Sentence can be expressed as$\forall x\left(P\left(x\right)\vee Q\left(x\right)\right)$

To Consist of all students in the class

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

1. The Universal Quantifier:$\forall xP\left(x\right)$–The statement is true for at least one element in the domain.
2. The Existential Quantifier:$\exists xP\left(x\right)$– 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 exists a student who speaks Russian and knows C++ this is a case of the existential quantifier.

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

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