Question

Find a compound proposition involving the propositional variables, $\mathit{p}\mathbf{,}\mathit{q}$and r that is true when exactly two of, $\mathit{p}\mathbf{,}\mathit{q}$and r are true and is false otherwise. [Hint: Form a disjunction of conjunctions. Include a conjunction for each combination of values for which the compound proposition is true. Each conjunction should include each of the three propositional variables or its negations.]

Short answer

$\mathbf{(}\mathit{p}\mathbf{\wedge }\mathit{q}\mathbf{\wedge }\mathbf{\neg }\mathit{r}\mathbf{)}\mathbf{\vee }\mathbf{(}\mathit{p}\mathbf{\wedge }\mathbf{\neg }\mathit{q}\mathbf{\wedge }\mathit{r}\mathbf{)}\mathbf{\vee }\mathbf{(}\mathbf{\neg }\mathit{p}\mathbf{\wedge }\mathit{q}\mathbf{\wedge }\mathit{r}\mathbf{)}$

Step-by-step solution

LOGICAL COMPOUND

The hint to this problem is the previous exercise. The list of compound statements in $\mathit{p}\mathbf{,}\mathit{q}\mathbf{,}\mathit{r}$which are true precisely when two of the variables are true is as follows:

$\mathit{p}\mathbf{\wedge }\mathit{q}\mathbf{\wedge }\mathbf{\neg }\mathit{r}$

$\mathit{p}\mathbf{\wedge }\mathbf{\neg }\mathit{q}\mathbf{\wedge }\mathit{r}$

$\mathbf{\neg }\mathit{p}\mathbf{\wedge }\mathit{q}\mathbf{\wedge }\mathit{r}$

If two of the variables are true and the other is false, precisely one of the three compound statements is true. Otherwise, each of the statements is false.

Join them with disjunction logical connective

Join them with disjunction logical connective V to obtain

$(p\wedge q\wedge \neg r)\vee (p\wedge \neg q\wedge r)\vee (\neg p\wedge q\wedge r)$

This is the required answer to the question.