Question

Suppose the domain of the propositional function P(x,y) consists of pairs x and y, where x is 1,2, or 3 and y is 1,2, or 3. Write out these propositions using disjunctions and conjunctions.

$$\begin{gathered} \mathbf{a}\mathbf{)}\mathbf{}\mathbf{\forall }\mathbf{x}\mathbf{\forall }\mathbf{yP}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{b}\mathbf{)}\mathbf{}\exists \mathbf{x}\exists \mathbf{yP}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)} \\ \mathbf{c}\mathbf{)}\mathbf{}\exists \mathbf{x}\mathbf{\forall }\mathbf{yP}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{d}\mathbf{)}\mathbf{\forall }\mathbf{y}\exists \mathbf{xP}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)} \end{gathered}$$

Short answer

Propositions using disjunctions and conjunctions can be found

Step-by-step solution

Finding the Truth value for ∀x∀yP(x,y)

The Statement “$\forall \mathrm{x}\forall \mathrm{yP}(\mathrm{x},\mathrm{y})$” and

The domain of Propositional function P(x,y) is x, y where x = 1,2,3 & Y = 1,2,3.

The Statement “$\forall \mathrm{x}\forall \mathrm{yP}(\mathrm{x},\mathrm{y})$”, x = 1,2,3 & y = 1,2,3

The Propositional are role="math" localid="1668605556213" $\mathrm{P}(1,1)\wedge \mathrm{P}(1,2)\wedge \mathrm{P}(1,3)\wedge \mathrm{P}(2,1)\wedge \mathrm{P}(2,2)\wedge \mathrm{P}(3,3)$

As a result, The Propositional of “$\forall \mathrm{x}\forall \mathrm{yP}(\mathrm{x},\mathrm{y})$” are$\mathrm{P}(1,1)\wedge \mathrm{P}(1,2)\wedge \mathrm{P}(1,3)\wedge \mathrm{P}(2,1)\wedge \mathrm{P}(2,2)\wedge \mathrm{P}(3,3)$

Finding the Truth value for ∃x∃yP(x,y)

The Statement "$\exists \mathrm{x}\exists \mathrm{yP}(\mathrm{x},\mathrm{y})$" and

The domain of Propositional function P(x,y) is x, y where x = 1,2,3 & y = 1,2,3.

The Statement “$\exists \mathrm{x}\exists \mathrm{yP}(\mathrm{x},\mathrm{y})$”, x = 1,2,3 & y = 1,2,3

The Propositional are $\mathrm{P}(1,1)\vee \mathrm{P}(1,2)\vee \mathrm{P}(1,3)\vee \mathrm{P}(2,1)\vee \mathrm{P}(2,2)\vee \mathrm{P}(3,3)$

As a result, The Propositional of “$\exists \mathrm{x}\exists \mathrm{yP}(\mathrm{x},\mathrm{y})$” are$\mathrm{P}(1,1)\vee \mathrm{P}(1,2)\vee \mathrm{P}(1,3)\vee \mathrm{P}(2,1)\vee \mathrm{P}(2,2)\vee \mathrm{P}(3,3)$

Finding the Truth value for ∃x∀yP(x,y)

The Statement “$\exists \mathrm{x}\forall \mathrm{yP}(\mathrm{x},\mathrm{y})$” and

The domain of Propositional function P(x,y) is x, y where x = 1,2,3 & y = 1,2,3.

The Statement “$\exists \mathrm{x}\forall \mathrm{yP}(\mathrm{x},\mathrm{y})$”, x = 1,2,3 & y = 1,2,3

The Propositional are $\mathrm{P}(1,1)\wedge \mathrm{P}(1,2)\wedge \mathrm{P}(1,3)\vee \mathrm{P}(2,1)\wedge \mathrm{P}(2,2)\wedge \mathrm{P}(3,3)$

As a result, The Propositional of “$\exists \mathrm{x}\forall \mathrm{yP}(\mathrm{x},\mathrm{y})$” are$\mathrm{P}(1,1)\wedge \mathrm{P}(1,2)\wedge \mathrm{P}(1,3)\vee \mathrm{P}(2,1)\wedge \mathrm{P}(2,2)\wedge \mathrm{P}(3,3)$

Finding the Truth value for ∀y∃xP(x,y)

The Statement “$\forall \mathrm{y}\exists \mathrm{xP}(\mathrm{x},\mathrm{y})$” and

The domain of Propositional function P(x,y) is x, y where x = 1,2,3 & Y = 1,2,3.

The Statement “$\forall \mathrm{y}\exists \mathrm{xP}(\mathrm{x},\mathrm{y})$”, x = 1,2,3 & Y = 1,2,3

The Propositional are$\left(\mathrm{P}\right(1,1)\vee \mathrm{P}(1,2)\vee \mathrm{P}(1,3\left)\right)\wedge \left(\mathrm{P}\right(2,1)\vee \mathrm{P}(2,2)\vee )(2,3))\wedge (\mathrm{P}(3,1)\vee \mathrm{P}(3,2)\vee \mathrm{P}(3,3))$

As a result, The Propositional of “$\forall \mathrm{y}\exists \mathrm{xP}(\mathrm{x},\mathrm{y})$” are$\left(\mathrm{P}\right(1,1)\vee \mathrm{P}(1,2)\vee \mathrm{P}(1,3\left)\right)\wedge \left(\mathrm{P}\right(2,1)\vee \mathrm{P}(2,2)\vee )(2,3))\wedge (\mathrm{P}(3,1)\vee \mathrm{P}(3,2)\vee \mathrm{P}(3,3))$