Question

Let P(x) be the statement “$\mathbf{x}\mathbf{=}{\mathbf{x}}^{\mathbf{2}}$.”If the domain consists of the integers, what are these truth values?

$$\begin{gathered} \mathbf{a}\mathbf{)}\mathbf{}\mathbf{P}\mathbf{(}\mathbf{0}\mathbf{\left)} \\ \mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{P}\mathbf{\left(}\mathbf{1}\mathbf{\right)} \\ \mathbf{c}\mathbf{)}\mathbf{}\mathbf{P}\mathbf{(}\mathbf{2}\mathbf{\left)} \\ \mathbf{d}\mathbf{\right)}\mathbf{}\mathbf{P}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)} \\ \mathbf{e}\mathbf{)}\mathbf{\exists }\mathbf{xP}\mathbf{(}\mathbf{x}\mathbf{\left)} \\ \mathbf{f}\mathbf{\right)}\mathbf{}\mathbf{\forall }\mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)} \end{gathered}$$

Short answer

a) The Truth value of $\mathrm{P}\left(0\right)$ is true.

b) The Truth value of $\mathrm{P}\left(1\right)$is true.

(c) The Truth value of $\mathrm{P}\left(2\right)$is False.

(d) The Truth value of width="46" height="21" role="math" style="max-width: none; vertical-align: -5px;" localid="1668495956284" $\mathrm{P}(-1)$ is False.

(e) The Truth value of$\exists \mathrm{xP}\left(\mathrm{x}\right)$ is true.

(f) The Truth value of $\forall \mathrm{xP}\left(\mathrm{x}\right)$is False.

Step-by-step solution

Given that variable consists of all integers

The Given that P(x) denoted as x=x² 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 P(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=x²

To find P(0), Substitute x=0

$\mathrm{x}=0$

${\mathrm{x}}^2={\left(0\right)}^2$

$=0$

$\Rightarrow 0=0$

Thus, P(0) is True

As a result, P(0) is true for the Statement x=x².

The Truth value of P(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=x²

To find P(1), Substitute x=1

x=1

x²=(1)²=1

$\Rightarrow 1=1$

Thus,P(1) is True

As a result,P(1) is true for the Statement x=x².

The Truth value of P(2)

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=x²

To find P(2), Substitute x=2

x=2

x²=(2)²=4

$\Rightarrow 2\ne 4$

Thus,P(2) is False

As a result, P(2) is False for the Statement x=x².

The Truth value of P(-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=x²

To find P(-1) , Substitute x= -1

$\mathrm{x}=-1$

${\mathrm{x}}^2={\left(-1\right)}^2=1$

$\Rightarrow 1\ne -1$

Thus, P(-1) is False

As a result, P(-1) is False for the Statement x=x².

The Truth value of ∃xP(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=x²

We have $\exists \mathrm{xP}\left(\mathrm{x}\right)$, the statement is$\exists \mathrm{xP}(\mathrm{x}={\mathrm{x}}^2)$

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

We have x=x²

For x=0

LHS = 0

RHS = (0)²=0

$\Rightarrow 0=0$

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

As a result, $\exists \mathrm{xP}\left(\mathrm{x}\right)$is true for the Statement $\mathrm{x}={\mathrm{x}}^2$.

The Truth value of ∀xP(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=x².

We have $\forall \mathrm{xP}\left(\mathrm{x}\right)$,the statement is$\forall \mathrm{xP}\left(\mathrm{x}={\mathrm{x}}^2\right)$

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

We have x=x²

For x=2

LHS = 2

RHS = (2)²=4

width="56" style="max-width: none;" localid="1668500704346" $\Rightarrow 2\ne 4$

Thus,The statement is False as it satisfies existential quantification.

As a result, $\forall \mathrm{xP}\left(\mathrm{x}\right)$is False for the Statement x=x².