Question

17 E. Suppose that the domain of the propositional function P(x) consists of the integers 0,1,2,3,4. Write out each of these propositions using disjunctions, conjunctions and negations.

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

Short answer

1. The propositional function$\exists \mathrm{xP}\left(\mathrm{x}\right)$is denoted asrole="math" localid="1668500932601" $\mathbf{P}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\vee \mathbf{P}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\vee \mathbf{P}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\vee \mathbf{P}\mathbf{\left(}\mathbf{3}\mathbf{\right)}\vee \mathbf{P}\mathbf{\left(}\mathbf{4}\mathbf{\right)}$.
2. The propositional function $\forall \mathrm{xP}\left(\mathrm{x}\right)$is denoted as role="math" localid="1668500964700" $\mathbf{P}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\wedge \mathbf{P}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\wedge \mathbf{P}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\wedge \mathbf{P}\mathbf{\left(}\mathbf{3}\mathbf{\right)}\wedge \mathbf{P}\mathbf{\left(}\mathbf{4}\mathbf{\right)}$.
3. The propositional function$\exists \mathrm{x}\neg \mathrm{P}\left(\mathrm{x}\right)$is denoted as role="math" localid="1668500754333" $\mathbf{\neg }\mathbf{\left[}\mathbf{P}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{\wedge }\mathbf{P}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{\wedge }\mathbf{P}\mathbf{\left(}\mathbf{3}\mathbf{\right)}\mathbf{\wedge }\mathbf{P}\mathbf{\left(}\mathbf{4}\mathbf{\right)}\mathbf{\wedge }\mathbf{P}\mathbf{\left(}\mathbf{5}\mathbf{\right)}\mathbf{\right]}$.
4. The propositional function$\forall \mathrm{x}\neg \mathrm{P}\left(\mathrm{x}\right)$is denoted as role="math" localid="1668500808744" $\mathbf{\neg }\mathbf{\left[}\mathbf{P}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\vee \mathbf{P}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\vee \mathbf{P}\mathbf{\left(}\mathbf{3}\mathbf{\right)}\vee \mathbf{P}\mathbf{\left(}\mathbf{4}\mathbf{\right)}\vee \mathbf{P}\mathbf{\left(}\mathbf{5}\mathbf{\right)}\mathbf{\right]}$.
5. The propositional function$\neg \exists \mathrm{xP}\left(\mathrm{x}\right)$is denoted as $\mathbf{\neg }\mathbf{\left[}\mathbf{P}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\vee \mathbf{P}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\vee \mathbf{P}\mathbf{\left(}\mathbf{3}\mathbf{\right)}\vee \mathbf{P}\mathbf{\left(}\mathbf{4}\mathbf{\right)}\vee \mathbf{P}\mathbf{\left(}\mathbf{5}\mathbf{\right)}\mathbf{\right]}$.
6. The propositional function$\neg \forall \mathrm{xP}\left(\mathrm{x}\right)$is denoted as $\mathbf{\neg }\mathbf{\left[}\mathbf{P}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{\wedge }\mathbf{P}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{\wedge }\mathbf{P}\mathbf{\left(}\mathbf{3}\mathbf{\right)}\mathbf{\wedge }\mathbf{P}\mathbf{\left(}\mathbf{4}\mathbf{\right)}\mathbf{\wedge }\mathbf{P}\mathbf{\left(}\mathbf{5}\mathbf{\right)}\mathbf{\right]}$.

Step-by-step solution

a. Defining Propositional function

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

Conjunctions of proposition given as $\mathrm{p}\wedge \mathrm{q}$. It is true when both p and q are true and is false otherwise.

Disjunction of proposition is given as $\mathrm{p}\vee \mathrm{q}$. It is false when both p and q are false and is true otherwise.

Negation of proposition given as $\neg \mathrm{p}$, is truth value and negation of is opposite of the true value p.

To write propositional function:

The domain of the propositional function contains integers 0,1,2,3,4.

Given propositional function is $\exists \mathrm{xP}\left(\mathrm{x}\right)$.

As the domain is {0,1,2,3,4}, the given proposition equals disjunction of domain.

Hence, $\exists \mathrm{xP}\left(\mathrm{x}\right)=\mathrm{P}\left(0\right)\vee \mathrm{P}\left(1\right)\vee \mathrm{P}\left(2\right)\vee \mathrm{P}\left(3\right)\vee \mathrm{P}\left(4\right)$.

b. Defining Propositional function

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

Conjunctions of proposition given as $\mathrm{p}\wedge \mathrm{q}$. It is true when both p and q are true and is false otherwise.

Disjunction of proposition is given as $\mathrm{p}\vee \mathrm{q}$. It is false when both p and q are false and is true otherwise.

Negation of proposition given as -p, is truth value is opposite of true value p.

To write propositional function:

The domain of the propositional function contains integers 0,1,2,3,4.

Given propositional function is $\forall \mathrm{xP}\left(\mathrm{x}\right)$.

As the domain is {0,1,2,3,4}, the given proposition equals conjunction of domain.

Hence, $\forall \mathrm{xP}\left(\mathrm{x}\right)=\mathrm{P}\left(0\right)\wedge \mathrm{P}\left(1\right)\wedge \mathrm{P}\left(2\right)\wedge \mathrm{P}\left(3\right)\wedge \mathrm{P}\left(4\right)$.

c. Defining Propositional function

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

Conjunctions of proposition given as $\mathrm{p}\wedge \mathrm{q}$. It is true when both p and q are true and is false otherwise.

Disjunction of proposition is given as $\mathrm{p}\vee \mathrm{q}$. It is false when both p and q are false and is true otherwise.

Negation of proposition given as -p, is truth value is opposite of true value p.

To write propositional function:

The domain of the propositional function contains integers 0,1,2,3,4.

Given propositional function is $\exists \mathrm{x}\neg \mathrm{P}\left(\mathrm{x}\right)$.

By applying De Morgan’s law, it is written as role="math" localid="1668502312693" $\exists \mathrm{x}\neg \mathrm{P}\left(\mathrm{x}\right)=\neg \forall \mathrm{xP}\left(\mathrm{x}\right)$

As the domain is {0,1,2,3,4}, the given proposition equals negation of conjunction of domain.

Hence, $\exists \mathrm{x}\neg \mathrm{P}\left(\mathrm{x}\right)=\neg \forall \mathrm{xP}\left(\mathrm{x}\right)=\neg \left[\mathrm{P}\left(1\right)\wedge \mathrm{P}\left(2\right)\wedge \mathrm{P}\left(3\right)\wedge \mathrm{P}\left(4\right)\wedge \mathrm{P}\left(5\right)\right]$.

d. Defining Propositional function

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

Conjunctions of proposition given as $\mathrm{p}\wedge \mathrm{q}$. It is true when both p and q are true and is false otherwise.

Disjunction of proposition is given as $\mathrm{p}\vee \mathrm{q}$. It is false when both p and q are false and is true otherwise.

Negation of proposition given as -p, is truth value is opposite of true value p.

To write propositional function:

The domain of the propositional function contains integers 0,1,2,3,4.

Given propositional function is $\forall x\neg P\left(\mathrm{x}\right)$.

By applying De Morgan’s law, it is written as $\forall \mathrm{x}\neg \mathrm{P}\left(\mathrm{x}\right)=\neg \exists \mathrm{xP}\left(\mathrm{x}\right)$

As the domain is {0,1,2,3,4}, the given proposition equals negation of disjunction of domain.

Hence, $\forall \mathrm{x}\neg \mathrm{P}\left(\mathrm{x}\right)=\neg \exists \mathrm{xP}\left(\mathrm{x}\right)=\neg \left[\mathrm{P}\left(1\right)\vee \mathrm{P}\left(2\right)\vee \mathrm{P}\left(3\right)\vee \mathrm{P}\left(4\right)\vee \mathrm{P}\left(5\right)\right]$.

e. Defining Propositional function

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

Conjunctions of proposition given as $\mathrm{p}\wedge \mathrm{q}$. It is true when both p and q are true and is false otherwise.

Disjunction of proposition is given as $\mathrm{p}\vee \mathrm{q}$. It is false when both p and q are false and is true otherwise.

Negation of proposition given as -p, is truth value is opposite of true value p.

To write propositional function:

The domain of the propositional function contains integers 0,1,2,3,4.

Given propositional function is $\neg \exists \mathrm{xP}\left(\mathrm{x}\right)$.

As the domain is {0,1,2,3,4}, the given proposition equals negation of disjunction of domain.

Hence, $\neg \exists \mathrm{xP}\left(\mathrm{x}\right)=\neg \left[\mathrm{P}\left(1\right)\vee \mathrm{P}\left(2\right)\vee \mathrm{P}\left(3\right)\vee \mathrm{P}4)\vee \mathrm{P}(5)\right]$.

f. Defining Propositional function

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

Conjunctions of proposition given as $\mathrm{p}\wedge \mathrm{q}$. It is true when both p and q are true and is false otherwise.

Disjunction of proposition is given as $\mathrm{p}\vee \mathrm{q}$. It is false when both p and q are false and is true otherwise.

Negation of proposition given as -p, is truth value is opposite of true value p.

To write propositional function:

The domain of the propositional function contains integers 0,1,2,3,4.

Given propositional function is $\neg \forall \mathrm{xP}\left(\mathrm{x}\right)$.

As the domain is {0,1,2,3,4}, the given proposition equals negation of conjunction of domain.

Hence, $\neg \forall \mathrm{xP}\left(\mathrm{x}\right)=\neg \left[\mathrm{P}\left(1\right)\wedge \mathrm{P}\left(2\right)\wedge \mathrm{P}\left(3\right)\wedge \mathrm{P}\left(4\right)\wedge \mathrm{P}\left(5\right)\right]$.