Question

Determine whether each of these compound propositions is satisfiable.

$$\begin{gathered} \mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{}\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q}\mathbf{)} \\ \mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{\neg }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\to }\mathbf{\neg }\mathbf{q}\mathbf{)} \\ \mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{}\mathbf{(}\mathbf{p}\mathbf{\leftrightarrow }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\leftrightarrow }\mathbf{q}\mathbf{)} \end{gathered}$$

Short answer

The compound proposition $\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q}\mathbf{)}$is satisfiable while the other compound propositions such as $\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{\neg }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\to }\mathbf{\neg }\mathbf{q}\mathbf{)}$and $\mathbf{(}\mathbf{p}\mathbf{\leftrightarrow }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\leftrightarrow }\mathbf{q}\mathbf{)}$ are unsatisfiable. Prepare a truth table for each proposition and check whether it is a satisfiable or not.

Step-by-step solution

Definition of the truth table  

A logic gate truth table depicts each feasible input sequence to the gate or circuit, as well as the resulting output based on the combination of these inputs.

To determine whether each of these compound propositions is satisfiable.

$$\begin{gathered} \mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{}\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q}\mathbf{)} \\ \mathrm{Prepare}\mathrm{the}\mathrm{truth}\mathrm{table}\mathrm{for}\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q}\mathbf{)} \end{gathered}$$

p	q	$\mathbf{\neg }\mathbf{p}$	$\mathbf{\neg }\mathbf{q}$	$\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q}$	localid="1668414422585" $\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{q}$	$\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q}$	Compound Propositions
T	T	F	F	T	T	F	F
T	F	F	T	T	F	T	F
F	T	T	F	F	T	T	F
F	F	T	T	T	T	T	T

Truth Table

In the above truth table, the compound proposition is true if p and q are false.

Therefore, the compound proposition$\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q}\mathbf{)}\mathbf{}$is satisfiable.

$$\begin{gathered} \mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{\neg }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\to }\mathbf{\neg }\mathbf{q}\mathbf{)} \\ Preparethetruthtablefor\mathbf{}\mathbf{}\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{\neg }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\to }\mathbf{\neg }\mathbf{q}\mathbf{)} \end{gathered}$$

p	q	$\mathbf{\neg }\mathbf{p}$	$\mathbf{\neg }\mathbf{q}$	$\mathbf{p}\mathbf{\to }\mathbf{q}$	$\mathbf{p}\mathbf{\to }\mathbf{\neg }\mathbf{q}$$\mathbf{p}\mathbf{\to }\mathbf{\neg }\mathbf{q}$	$\mathbf{\neg }\mathbf{p}\mathbf{\to }\mathbf{q}$	$\mathbf{\neg }\mathbf{p}\mathbf{\leftrightarrow }\mathbf{\neg }\mathbf{q}$	Compound Proposition
T	T	F	F	T	F	T	T	F
T	F	F	T	F	T	T	T	F
F	T	T	F	T	T	T	F	F
F	F	T	T	T	T	F	T	F

Truth Table

In the above truth table, the compound proposition is always false.

Therefore, the compound proposition $\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{\neg }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\to }\mathbf{\neg }\mathbf{q}\mathbf{)}$ is unsatisfiable.

(c)$\mathbf{(}\mathbf{p}\mathbf{\leftrightarrow }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\leftrightarrow }\mathbf{q}\mathbf{)}$

Prepare the truth table for $\mathbf{(}\mathbf{p}\mathbf{\leftrightarrow }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\leftrightarrow }\mathbf{q}\mathbf{)}$

p	q	$\mathbf{\neg }\mathbf{p}$	$\mathbf{(}\mathbf{p}\mathbf{\leftrightarrow }\mathbf{q}\mathbf{)}$	$\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\leftrightarrow }\mathbf{q}\mathbf{)}$	Compound Proposition
T	T	F	T	F	F
T	F	F	F	T	F
F	T	T	F	T	F
F	F	T	T	F	F

Truth Table

In the above truth table, the compound proposition is always false.

Therefore, the compound proposition $\mathbf{(}\mathbf{p}\mathbf{\leftrightarrow }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{\leftrightarrow }\mathbf{q}\mathbf{)}$is unsatisfiable.