As the provided condition is, the compound proposition involving variables\(p,q,r,and\,s\)is true when three propositional variables are true and is false otherwise.
\(\left( {p \wedge q \wedge r \wedge \neg s} \right) \vee \left( {p \wedge q \wedge \neg r \wedge s} \right) \vee \left( {p \wedge \neg q \wedge r \wedge s} \right) \vee \left( {\neg p \wedge q \wedge r \wedge s} \right)\)
Now, it is assumed that,
\(\begin{aligned}{}P = \left( {p \wedge q \wedge r \wedge \neg s} \right)\\Q = \left( {p \wedge q \wedge \neg r \wedge s} \right)\\R = \left( {p \wedge \neg q \wedge r \wedge s} \right)\\S = \left( {\neg p \wedge q \wedge r \wedge s} \right)\\W = \left( {p \wedge q \wedge r \wedge \neg s} \right) \vee \left( {p \wedge q \wedge \neg r \wedge s} \right) \vee \left( {p \wedge \neg q \wedge r \wedge s} \right) \vee \left( {\neg p \wedge q \wedge r \wedge s} \right)\end{aligned}\)
Now the truth table is as shown below;
\(p\) | \(q\) | \(r\) | \(s\) | \(P\) | \(Q\) | \(R\) | \(S\) | \(W\) |
F | F | F | F | F | F | F | F | F |
F | F | F | T | F | F | F | F | F |
F | F | T | F | F | F | F | F | F |
F | F | T | T | F | F | F | F | F |
F | T | F | F | F | F | F | F | F |
F | T | F | T | F | F | F | F | F |
F | T | T | F | F | F | F | F | F |
F | T | T | T | F | F | F | T | T |
T | F | F | F | F | F | F | F | F |
T | F | F | T | F | F | F | F | F |
T | F | T | F | F | F | F | F | F |
T | F | T | T | F | F | T | F | T |
T | T | F | F | F | F | F | F | F |
T | T | F | T | F | T | F | F | T |
T | T | T | F | T | F | F | F | T |
T | T | T | T | F | F | F | F | F |
So proposition is
\(\left( {p \wedge q \wedge r \wedge \neg s} \right) \vee \left( {p \wedge q \wedge \neg r \wedge s} \right) \vee \left( {p \wedge \neg q \wedge r \wedge s} \right) \vee \left( {\neg p \wedge q \wedge r \wedge s} \right)\).
