Question

19E. Suppose that the domain of the propositional function consists of the integers 1,2,3,4,5. Express these statements without using the quantifiers, instead using disjunctions, conjunctions and negations.

$$\begin{gathered} \mathbf{a}\mathbf{)}\mathbf{}\exists \mathbf{xP}\mathbf{(}\mathbf{x}\mathbf{\left)} \\ \mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{\forall }\mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)} \\ \mathbf{c}\mathbf{)}\mathbf{}\exists \mathbf{x}\mathbf{\neg }\mathbf{P}\mathbf{(}\mathbf{x}\mathbf{\left)} \\ \mathbf{d}\mathbf{\right)}\mathbf{}\mathbf{\forall }\mathbf{x}\mathbf{\neg }\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)} \\ \mathbf{e}\mathbf{)}\mathbf{}\mathbf{\forall }\mathbf{x}\mathbf{(}\mathbf{x}\mathbf{\ne }\mathbf{3}\mathbf{)}\mathbf{\to }\exists \mathbf{x}\mathbf{\neg }\mathbf{P}\mathbf{(}\mathbf{x}\mathbf{)} \end{gathered}$$

Short answer

1. The propositional function$\exists \mathrm{xP}\left(\mathrm{x}\right)$is denoted as role="math" localid="1668489506970" $\mathbf{P}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{\vee }\mathbf{P}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{\vee }\mathbf{P}\mathbf{\left(}\mathbf{3}\mathbf{\right)}\mathbf{\vee }\mathbf{P}\mathbf{\left(}\mathbf{4}\mathbf{\right)}\mathbf{\vee }\mathbf{P}\mathbf{\left(}\mathbf{5}\mathbf{\right)}$.
2. The propositional function$\forall \mathrm{xP}\left(\mathrm{x}\right)$is denoted asrole="math" localid="1668489705801" $\mathbf{P}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{\wedge }\mathbf{P}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{\wedge }\mathbf{P}\mathbf{\left(}\mathbf{3}\mathbf{\right)}\mathbf{\wedge }\mathbf{P}\mathbf{\left(}\mathbf{4}\mathbf{\right)}\mathbf{\wedge }\mathbf{P}\mathbf{\left(}\mathbf{5}\mathbf{\right)}$.
3. The propositional function $\exists \mathrm{x}\neg \mathrm{P}\left(\mathrm{x}\right)$is denoted as $\mathbf{\neg }\left[P\left(1\right)\vee P\left(2\right)\vee P\left(3\right)\vee P\left(4\right)\vee P\left(5\right)\right]$
   
4. The propositional function $\forall \mathrm{x}\neg \mathrm{P}\left(\mathrm{x}\right)$is denoted as $\mathbf{\neg }\left[P\left(1\right)\wedge P\left(2\right)\wedge P\left(3\right)\wedge P\left(4\right)\wedge P\left(5\right)\right]$.
5. The propositional function$\forall \mathbf{xP}\mathbf{\left(}\mathbf{\right(}\mathbf{x}\mathbf{\ne }\mathbf{3}\mathbf{)}\mathbf{\to }\mathbf{P}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\vee \exists \mathbf{x}\mathbf{\neg }\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$ is denoted as$\left[P\left(1\right)\mathrm{\wedge }P\left(2\right)\mathrm{\wedge }P\left(4\right)\mathrm{\wedge }P\left(5\right)\right]$$\vee \left[\neg P\left(1\right)\mathrm{\vee }\neg P\left(2\right)\mathrm{\vee }\neg P\left(3\right)\mathrm{\vee }\neg P\left(4\right)\mathrm{\vee }\neg P\left(5\right)\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 1,2,3,4,5

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

As the domain is $\left\{1,2,3,4,5\right\}$, the given proposition equals disjunction of domain.

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

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 and p are q 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 1,2,3,4,5.

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

As the domain is $\left\{1,2,3,4,5\right\}$, the given proposition equals conjunction of domain.

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

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 1,2,3,4,5.

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 $\left\{1,2,3,4,5\right\}$, the given proposition equals negation of conjunction of domain.

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

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 1,2,3,4,5.

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 $\left\{1,2,3,4,5\right\}$, 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 1,2,3,4,5.

Given propositional function is $\forall \mathrm{xP}\left(\right(\mathrm{x}\ne 3)\to (\mathrm{P}\left(\mathrm{x}\right))\vee \exists \mathrm{x}\neg \mathrm{p}(\mathrm{x})$.

As the domain is $\left\{1,2,3,4,5\right\}$, the given proposition equals conjunction of disjunction of domain for value except x = 3, it is conjunction of negation of domain.

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