Question

Letbe the statement “x+1>2x.”If the domain consists of the integers, what are these truth values?

$$\begin{gathered} \mathbf{a}\mathbf{)}\mathbf{}\mathbf{Q}\mathbf{(}\mathbf{0}\mathbf{)}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{b}\mathbf{)}\mathbf{}\mathbf{Q}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{c}\mathbf{)}\mathbf{}\mathbf{Q}\mathbf{(}\mathbf{1}\mathbf{\left)} \\ \mathbf{d}\mathbf{\right)}\mathbf{}\mathbf{\exists }\mathbf{xQ}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{e}\mathbf{)}\mathbf{}\mathbf{\forall }\mathbf{xQ}\mathbf{(}\mathbf{x}\mathbf{)}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{f}\mathbf{)}\mathbf{}\mathbf{\exists }\mathbf{xQ}\mathbf{\neg }\mathbf{\left(}\mathbf{x}\mathbf{\right)} \\ \mathbf{g}\mathbf{)}\mathbf{}\mathbf{\forall }\mathbf{x}\mathbf{\neg }\mathbf{Q}\mathbf{(}\mathbf{x}\mathbf{)} \end{gathered}$$

Short answer

(a) The Truth value of Q(0) is true.

(b) The Truth value of $\mathrm{Q}(-1)$is true.

(c) The Truth value of $\mathrm{Q}\left(1\right)$is False.

(d) The Truth value of $\exists \mathrm{xQ}\left(\mathrm{x}\right)$is True.

(e) The Truth value of $\forall \mathrm{xQ}\left(\mathrm{x}\right)$is True.

(f) The Truth value of $\exists \mathrm{xQ}\neg \left(\mathrm{x}\right)$is True.

(g) The Truth value of $\forall \mathrm{x}\neg \mathrm{Q}\left(\mathrm{x}\right)$is False.

Step-by-step solution

Given

The Given that Q(x) denoted as x+1>2x and the domain of the variable consists of all integers.

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.

The Truth value of Q(0)

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 thatdomain of the variable is all integers

Also, The statement is x+1>2x

Q(0)=0+1>2.0

$\Rightarrow 1>0$

Thus, Q(0) is True

As a result, Q(0) is true for the Statement x+1>2x.

The Truth value of  Q(-1)

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 domain of the variable is all integers

Also, The statement is x+1>2x

Q(-1) = (-1) +1>2(-1)

$\Rightarrow 0$>-2

Thus, Q(-1) is True

As a result, Q(-1) is true for the Statement x+1>2x.

The Truth value of  Q(1)

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 domain of the variable is all integers

Also, The statement is x+1>2x

Q(1)=(1) + 1>2.(1)

$\Rightarrow 2>2$

Thus, Q(1) is False as 2=2

As a result, Q(1) is False for the Statement x+1>2x.

The Truth value of ∃xQ(x)

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 domain of the variable is all integers

Also, The statement is x+1 > 2x

We have $\exists \mathrm{xQ}\left(\mathrm{x}\right)$, the statement is$\exists \mathrm{x}(\mathrm{x}+1>2\mathrm{x})$

This is existential quantifier, i.e. there exists at least one element for which the statement is true

We have x+1>2x

For x=0

LHS=0+1=1

RHS=2.0=0

1>0

Thus, the statement is true as it satisfies existential quantification.

As a result, $\exists \mathrm{xQ}\left(\mathrm{x}\right)$ is True for the Statement x+1>2x .

The Truth value of ∀xQ(x)

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 domain of the variable is all integers

Also, The statement is x+1>2x

We have $\forall \mathrm{xP}\left(\mathrm{x}\right)$, the statement isrole="math" localid="1668520032217" $\forall \mathrm{xP}(\mathrm{x}+1>2\mathrm{x})$

This is universal quantifier, i.e. the statement is true for all values in domain

We have x+1>2x

For x=1

LHS=1+1=2

RHS=2.1=2

$\Rightarrow 2=2$

Thus,The statement is False as it does not satisfy universal quantification.

As a result, $\forall \mathrm{xQ}\left(\mathrm{x}\right)$ is true for the Statement x+1>2x.

The Truth value of ∃x¬Q(x)

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 domain of the variable is all integers

Also, The statement is x+1 >2x

We have $\exists \mathrm{x}\neg \mathrm{Q}\left(\mathrm{x}\right)$,the statement is$\exists \mathrm{x}\neg \mathrm{Q}(\mathrm{x}+1>2\mathrm{x})$

This is existential quantifier, i.e. there exists at least one element for which the statement is true

We have x+1>2x

For x=1

LHS=1+1=2

RHS=2.1=2

$\Rightarrow 2=2$

Hence, x+1 >2x

Thus, the statement is True as it satisfies existential quantification.

As a result, $\exists \mathrm{x}\neg \mathrm{Q}\left(\mathrm{x}\right)$ is True for the Statement x+1 >2x.

The Truth value of ∀x¬Q(x)

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 domain of the variable is all integers

Also, The statement is x+1>2x

We have $\forall \mathrm{x}\neg \mathrm{Q}\left(\mathrm{x}\right)$,the statement is$\forall \mathrm{x}\neg \mathrm{Q}(\mathrm{x}+1>2\mathrm{x})$

This is existential quantifier, i.e. there exists at least one element for which the statement is true

We have x+1>2x

For x=0

LHS=1+0=1

RHS=2.0=0

$\Rightarrow 1>0$'
Thus, the statement is False as it does not satisfy universal quantification.

As a result, $\forall \mathrm{x}\neg \mathrm{Q}\left(\mathrm{x}\right)$ is True for the Statement x+1>2x.