Answer
The Required compound proposition involving the propositional
variables p, q, r, and s that is true when exactly three of
these propositional variables are true and is false otherwise is given below
(¬p∧q∧r∧s)∨(p∧¬q∧r∧s)∨(p∧q∧¬r∧s)∨(p∧q∧r∧¬s)
Work Step by Step
We have for variable $p$, $q$, $r$ and $s$. three of the variables are true and the other is false. However, we don't know which variable is false variable.
Case 1: $p$ is false. Then the other variables $q$,$ r$, $s$ are true and $¬p$ is true.
$\neg p \wedge q \wedge r \wedge s$
Case 2: $q$ is false. Then the other variables $p$,$ r$, $s$ are true and $¬q$ is true.
$p \wedge \neg q \wedge r \wedge s$
Case 3:$ r$ is false. Then the other variables $p$,$ q$, $s$ are true and $¬r$ is true.
$p \wedge q \wedge \neg r \wedge s$
Case 4: $s$ is false. Then the other variables $p$,$ q$,$ r$ are true and $¬s$ is true.
$p \wedge q \wedge r \wedge \neg s$
Conclusion: one of the $4$ cases has to be true,
thus case $1$ is true or case $2$ is true or case $3$ is true or case $4$ is true.
This means that we need to take the conjunction of the propositions in each case.
$(\neg p \wedge q \wedge r \wedge s) \vee(p \wedge \neg q \wedge r \wedge s) \vee(p \wedge q \wedge \neg r \wedge s) \vee(p \wedge q \wedge r \wedge \neg s)$
