Question

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

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

Short answer

1. The propositional function$\exists \mathrm{xP}\left(\mathrm{x}\right)$is denoted asrole="math" localid="1668495003697" $\mathbf{P}\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{)}\vee \mathbf{P}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}\vee \mathbf{P}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\vee \mathbf{P}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\vee \mathbf{P}\mathbf{\left(}\mathbf{2}\mathbf{\right)}$.
2. The propositional function $\forall \mathrm{xP}\left(\mathrm{x}\right)$is denoted as role="math" localid="1668495060406" $\mathbf{P}\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{)}\wedge \mathbf{P}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}\wedge \mathbf{P}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\wedge \mathbf{P}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\wedge \mathbf{P}\mathbf{\left(}\mathbf{2}\mathbf{\right)}$.
3. The propositional function$\exists \mathrm{x}\neg \mathrm{P}\left(\mathrm{x}\right)$is denoted asrole="math" localid="1668495460850" $\mathbf{\neg }\mathbf{\left[}\mathbf{P}\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{)}\wedge \mathbf{P}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}\wedge \mathbf{P}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\wedge \mathbf{P}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\wedge \mathbf{P}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{\right]}$.
4. The propositional function $\forall \mathrm{x}\neg \mathrm{P}\left(\mathrm{x}\right)$is denoted asrole="math" localid="1668495357843" $\mathbf{\neg }\mathbf{\left[}\mathbf{P}\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{)}\vee \mathbf{P}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}\vee \mathbf{P}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\vee \mathbf{P}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\vee \mathbf{P}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{\right]}$.
5. The propositional function $\neg \exists \mathrm{xP}\left(\mathrm{x}\right)$is denoted as $\mathbf{\neg }\mathbf{\left[}\mathbf{P}\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{)}\vee \mathbf{P}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}\vee \mathbf{P}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\vee \mathbf{P}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\vee \mathbf{P}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{\right]}$.
6. The propositional function $\neg \forall \mathrm{xP}\left(\mathrm{x}\right)$is denoted as $\mathbf{\neg }\mathbf{\left[}\mathbf{P}\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{)}\wedge \mathbf{P}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}\wedge \mathbf{P}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\wedge \mathbf{P}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\wedge \mathbf{P}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{\right]}$.

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 -2,-1,01,2.

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

As the domain is $\left\{-2,-1,0,1,2\right\}$, the given proposition equals disjunction of domain.

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

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 -2,-1,0,1,2.

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

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

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

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 -2,-1,0,1,2.

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

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

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

Hence, $\exists \mathrm{x}\neg \mathrm{P}\left(\mathrm{x}\right)=\neg \forall \mathrm{xP}\left(\mathrm{x}\right)=\neg \left[\mathrm{P}(-2)\wedge \mathrm{P}(-1)\wedge \mathrm{P}\left(0\right)\wedge \mathrm{P}\left(1\right)\wedge \mathrm{P}\left(2\right)\right]$.

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 -2,-1,0,1,2.

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

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 {-2,-1,0,1,2}, the given proposition equals negation of disjunction of domain.

Hence, role="math" localid="1668498580856" $\forall \mathrm{x}\neg \mathrm{P}\left(\mathrm{x}\right)=\neg \exists \mathrm{xP}\left(\mathrm{x}\right)=\neg \left[\mathrm{P}(-2)\vee \mathrm{P}(-1)\vee \mathrm{P}\left(0\right)\vee \mathrm{P}\left(1\right)\vee \mathrm{P}\left(2\right)\right]$.

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 -2,-1,0,1,2.

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

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

Hence, $\neg \exists \mathrm{xP}\left(\mathrm{x}\right)=\neg \left[\mathrm{P}(-2)\vee \mathrm{P}(-1)\vee \mathrm{P}\left(0\right)\vee \mathrm{P}\left(1\right)\vee \mathrm{P}\left(2\right)\right]$.

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 -2,-1,0,1,2.

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

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

Hence, $\neg \forall \mathrm{xP}\left(\mathrm{x}\right)=\neg \left[\mathrm{P}(-2)\wedge \mathrm{P}(-1)\wedge \mathrm{P}\left(0\right)\wedge \mathrm{P}\left(1\right)\wedge \mathrm{P}\left(2\right)\right]$.