Question

Determine the truth value of each of these statements if the domain consists of all integers.

$$\begin{gathered} \mathbf{a}\mathbf{)}\mathbf{}\mathbf{\forall }\mathbf{n}\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{>}\mathbf{n}\mathbf{)}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{b}\mathbf{)}\mathbf{\exists }\mathbf{n}\mathbf{(}\mathbf{2}\mathbf{n}\mathbf{=}\mathbf{3}\mathbf{n}\mathbf{)} \\ \mathbf{c}\mathbf{)}\mathbf{}\mathbf{\exists }\mathbf{n}\mathbf{(}\mathbf{n}\mathbf{=}\mathbf{-}\mathbf{n}\mathbf{)}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{d}\mathbf{)}\mathbf{}\mathbf{\forall }\mathbf{n}\left(3n\le 4n\right)\mathbf{} \end{gathered}$$

Short answer

(a) The Statement $\forall \mathrm{n}(\mathrm{n}+1>\mathrm{n})$is true.

(b) The Statement$\exists \mathrm{n}(2\mathrm{n}=3\mathrm{n})$ is true.

(c) The Statement$\exists \mathrm{n}(\mathrm{n}=-\mathrm{n})$ is true.

(d) The Statement$\forall \mathrm{n}(3\mathrm{n}\le 4\mathrm{n})$ is False.

Step-by-step solution

Given

The Given that $\forall \mathrm{n}(\mathrm{n}+1>\mathrm{n})$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 ∀n(n+1>n)

(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$\forall \mathrm{n}(\mathrm{n}+1>\mathrm{n})$

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

We have$\mathrm{n}+1>\mathrm{n}$

We are adding 1 to integer, thus$\mathrm{n}+1$ will always be greater than n.

Thus, the given statement is true as it satisfies universal quantification.

As a result, The Statement$\forall \mathrm{n}\left(\mathrm{n}+1>\mathrm{n}\right)$ is true.

The Truth value of ∃n(2n=3n)

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$\exists \mathrm{n}(2\mathrm{n}=3\mathrm{n})$

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

We have $2\mathrm{n}=3\mathrm{n}$

For$\mathrm{n}=0$

$2.0=3.0$

Thus, the given statement is true as it satisfies universal quantification.

As a result, The Statement$\exists \mathrm{n}(2\mathrm{n}=3\mathrm{n})$ is true.

The Truth value of ∃n(n=-n)

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$\exists \mathrm{n}(\mathrm{n}=-\mathrm{n})$

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

We have $(\mathrm{n}=-\mathrm{n})$

For $\mathrm{n}=0$,

$0=-0$

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

As a result, The Statement $\exists \mathrm{n}(\mathrm{n}=-\mathrm{n})$is true.

The Truth value of ∀n(3n≤4n)

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$\forall \mathrm{n}\left(3\mathrm{n}\le 4\mathrm{n}\right)$

This is Universal quantifier, i.e. the statement is true for all value in domain

We have$3\mathrm{n}\le 4\mathrm{n}$

For Positive real number,$3\mathrm{n}\le 4\mathrm{n}$

For Negative real number,role="math" localid="1668516907673" $-3\mathrm{n}\le -4\mathrm{n}$

For zero,$3.0=4.0$

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

As a result, The Statement$\forall \mathrm{n}(3\mathrm{n}\le 4\mathrm{n})$ is False.