Question

16E. Determine the truth value for each of these statement if the domain of each variable consists of all real numbers.

$$\begin{gathered} \mathbf{a}\mathbf{)}\mathbf{}\exists \mathbf{x}\mathbf{(}{\mathbf{x}}^{\mathbf{2}}\mathbf{=}\mathbf{2}\mathbf{\left)} \\ \mathbf{b}\mathbf{\right)}\mathbf{}\exists \mathbf{x}\mathbf{(}{\mathbf{x}}^{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{)} \\ \mathbf{c}\mathbf{)}\mathbf{}\mathbf{\forall }\mathbf{x}\mathbf{(}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathbf{\ge }\mathbf{1}\mathbf{\left)} \\ \mathbf{d}\mathbf{\right)}\mathbf{}\mathbf{\forall }\mathbf{x}\mathbf{(}{\mathbf{x}}^{\mathbf{2}}\mathbf{\ne }\mathbf{x}\mathbf{)} \end{gathered}$$

Short answer

1. The statement $\exists \mathrm{x}({\mathrm{x}}^2=2)$ is true.
2. The statement $\exists \mathrm{x}({\mathrm{x}}^2=-1)$ is false.
3. The statement $\forall \mathrm{x}({\mathrm{x}}^2+2\ge 1)$is true.
4. The statement $\forall \mathrm{x}({\mathrm{x}}^2\ne \mathrm{x})$is false.

Step-by-step solution

a. Defining quantifiers

Statement P(x) becomes propositional when x is assigned a truth value.

Two types of quantifiers are the universal quantifier and the existential quantifier.

1. The Existential Quantifier : $\forall \mathrm{xP}\left(\mathrm{x}\right)$- The statement is true when at least one element in domain exists.
2. The Universal Quantifier: $\exists \mathrm{xP}\left(\mathrm{x}\right)$- The statement is true for all values in domain.

Validating quantification

The given domain of the variables are all real numbers.

The Given statement is $\exists \mathrm{x}({\mathrm{x}}^2=2)$

Given, x² = 2

$$\begin{gathered} \mathrm{x}=\sqrt{2} \\ {\left(\sqrt{2}\right)}^2=2 \end{gathered}$$

The real number $\mathrm{x}=\sqrt{2}$gives x²= 2

Existential Quantification - statement is true when at least one element in domain exists.

Hence the given statement is true as it satisfies existential quantification.

b. Defining quantifiers

Statement P(x) becomes propositional whenxis assigned a truth value.

Two types of quantifiers are the universal quantifier and the existential quantifier.

1. The Existential Quantifier : $\forall \mathrm{xP}\left(\mathrm{x}\right)$- The statement is true when at least one element in domain exists.
2. The Universal Quantifier: $\exists \mathrm{xP}\left(\mathrm{x}\right)$- The statement is true for all values in domain.

Validating quantification

The given domain of the variables are all real numbers.

The Given statement is $\exists \mathrm{x}({\mathrm{x}}^2=-1)$

Given,${\mathrm{x}}^2=-1$

role="math" localid="1668509454499" ${\mathrm{x}}^2=\sqrt{-1}$

${\mathrm{x}}^2=\sqrt{-1}$is an imaginary number, so no real number exists.

Universal quantification, statement is true for all values in domain.

Hence the given statement is false as it does not satisfy universal quantification.

c. Defining quantifiers

Statement P(x) becomes propositional whenxis assigned a truth value.

Two types of quantifiers are the universal quantifier and the existential quantifier.

1. The Existential Quantifier : $\forall \mathrm{xP}\left(\mathrm{x}\right)$- The statement is true when at least one element in domain exists.
2. The Universal Quantifier: $\exists \mathrm{xP}\left(\mathrm{x}\right)$- The statement is true for all values in domain.

Validating quantification

The given domains of the variables are all real numbers.

The Given statement is$\forall \mathrm{x}({\mathrm{x}}^2+2\ge 1)$

Given, role="math" localid="1668509882473" ${\mathrm{x}}^2+2\ge 1$

$$\begin{gathered} {\mathrm{x}}^2+2\ge 0 \\ {\mathrm{x}}^2+1\ge 0 \end{gathered}$$

x² is positive as square values of any number is always positive.

Universal quantification, statement is true for all values in domain.

Hence the given statement is true as it satisfy universal quantification.

d. Defining quantifiers

Statement P(x) becomes propositional whenxis assigned a truth value.

Two types of quantifiers are the universal quantifier and the existential quantifier.

1. The Existential Quantifier : $\forall \mathrm{xP}\left(\mathrm{x}\right)$- The statement is true when at least one element in domain exists.
2. The Universal Quantifier: $\exists \mathrm{xP}\left(\mathrm{x}\right)$- The statement is true for all values in domain.

Validating quantification

The given domain of the variables are all real numbers.

The Given statement is Ɐx(x²≠ x)

Given,x²≠ x

If x = 1,

x = 1,(1)²= 1

Universal quantification, statement is true for all values in domain.

Hence the given statement is false as it does not satisfy universal quantification.