Question

15E. Determine the truth value of each of these statements if the domain for all variables consists of all integers.

$$\begin{gathered} \mathbf{a}\mathbf{)}\mathbf{}\mathbf{}\mathbf{}\mathbf{\forall }\mathbf{n}\mathbf{(}{\mathbf{n}}^{\mathbf{2}}\mathbf{\ge }\mathbf{0}\mathbf{\left)} \\ \mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{}\mathbf{}\exists \mathbf{n}\mathbf{(}{\mathbf{n}}^{\mathbf{2}}\mathbf{=}\mathbf{2}\mathbf{)} \\ \mathbf{c}\mathbf{)}\mathbf{}\mathbf{}\mathbf{}\mathbf{\forall }\mathbf{n}\mathbf{(}{\mathbf{n}}^{\mathbf{2}}\mathbf{\ge }\mathbf{n}\mathbf{\left)} \\ \mathbf{d}\mathbf{\right)}\mathbf{}\mathbf{}\mathbf{}\exists \mathbf{n}\mathbf{(}{\mathbf{n}}^{\mathbf{2}}\mathbf{<}\mathbf{0}\mathbf{)} \end{gathered}$$

Short answer

1. The given statement $\exists \mathrm{n}({\mathrm{n}}^2\ge 0)$is true.
2. The given statement $\exists \mathrm{n}({\mathrm{n}}^2=2)$is false.
3. The given statement $\forall \mathrm{n}({\mathrm{n}}^2\ge \mathrm{n})$is true.
4. The given statement $\exists \mathrm{n}({\mathrm{n}}^2<0)$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 atleast 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 integers.

The given statement is$\exists \mathrm{n}({\mathrm{n}}^2\ge 0)$

Given, ${\mathrm{n}}^2\ge 0$

For all the positive integers, n² is positive. So, n²> 0.

For all the positive integers, n² is positive. So, n²> 0 .

For 0, n² is positive. So, n²> 0 .

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

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

b. 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 atleast 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 integers.

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

Given, n²= 2

There is no integer whose square value is 2.

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

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

c. 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 atleast 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 integers.

The Given statement is $\forall \mathrm{n}({\mathrm{n}}^2\ge \mathrm{n})$

Given, ${\mathrm{n}}^2\ge \mathrm{n}$

For all the positive integers, n² is positive and greater than . So, n² > n.

For all the positive integers, n² is positive and greater than . So, n² > -n .

For 0, n² is positive. So, n²= n .

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

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

d. 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 atleast 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 integers.

The Given statement is$\exists \mathrm{n}({\mathrm{n}}^2<0)$

Given, n²< 0,

There is no integer whose square value is less than zero.

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

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