Question

Let p, q, and r be the propositions

p : You have the flu.

q: You miss the final examination.

r : You pass the course.

Express each of these propositions as an English sentence.

a) $\mathbf{p}\mathbf{\to }\mathbf{q}$

b) $\mathbf{\neg }\mathbf{q}\mathbf{\leftrightarrow }\mathbf{r}$

c) $\mathbf{q}\mathbf{\to }\mathbf{\neg }\mathbf{r}$

d)$\mathbf{p}\vee \mathbf{q}\vee \mathbf{r}$

e)$\mathbf{\left(}\mathbf{p}\mathbf{\to }\mathbf{\neg }\mathbf{r}\mathbf{\right)}\mathbf{\vee }\mathbf{\left(}\mathbf{q}\mathbf{\to }\mathbf{\neg }\mathbf{r}\mathbf{\right)}$

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

Short answer

1. The English sentence is, if you have the flu, then you miss the final examination.
2. The English sentence is,you do not miss the final examination if and only if you pass the course.
3. The English sentence is, If you miss the final examination then you do not pass the course.
4. The English sentence is, you have the flu or you miss the final examination or you pass the course.
5. The English sentence is, if you miss the final examination or you pass the course.
6. The English sentence is, you have the flu and you miss the final examination, you do not miss the final examination and you pass the course.

Step-by-step solution

(a) p→q

Consider the following proposition:

$p\to q$

Express to the given proposition as an English sentence.

Recall that $p\to q$is equivalent to if role="math" localid="1663697512795" $p$, then role="math" localid="1663697530514" $q$.

role="math" localid="1663697462289" $p\to q$: If you have the flu, then you miss the final examination.

Therefore, English sentence of the proposition role="math" localid="1663697486889" $p\to q$is, if you have the flu, then you miss the final examination.

(b) ¬q↔r

Consider the following proposition:

$\neg q\leftrightarrow r$

Express to the given proposition as an English sentence.

Recall that role="math" localid="1663698002236" $p\to q$is equivalent to $p$if and only if $q$and $\neg q$is equivalent to not $q$.

$\neg q$: You do not miss the final examination.

$\neg q\leftrightarrow r$: You do not miss the final examination if and only if you pass the course.

Therefore, English sentence of the proposition $\neg q\leftrightarrow r$is, you do not miss the final examination if and only if you pass the course.

(c) q→¬r

Consider the following proposition:

$q\to \neg r$

Express to the given proposition as an English sentence.

Recall the $p\to q$is equivalent to if $p$, then $q$and $\neg r$is equivalent to not r.

$\neg r$: you do not pass the course.

$q\to \neg r$: if you miss the final examination, then you do not pass the course.

Therefore, English sentence of the proposition $q\to \neg r$is,

If you miss the final examination then you do not pass the course.

(d) p∨q∨r

Consider the following proposition:

$p\vee q\vee r$

Express to the given proposition as an English sentence.

Recall that $p\vee q$is equivalent to $p$or$q$.

$p\vee q$: you have the flu or you miss the final examination.

$p\vee q\vee r$: you have the flu or miss the final examination or you pass the course.

Therefore, English sentence of the proposition $p\vee q\vee r$is

You have the flu or you miss the final examination or you pass the course.

(e) (p→¬r)∨(q→¬r)

Consider the following proposition:

$\left(p\to q\neg r\right)\vee \left(q\to \neg r\right)$

Express to the given proposition as an English sentence.

Recall that $p\to q$is equivalent to if $p$, then $q$and $\neg r$is equivalent to not r.

Recall that $p\vee q$is equivalent $p$or $q$.

$\neg r$:you do not pass the course.

$p\to \neg r$: if you have the flu, you do not pass the pass the course.

$q\to \neg r$: if you miss the final examination, then you do not pass thecourse.

role="math" localid="1663698789001" $\left(p\to \neg r\right)\vee \left(q\to \neg r\right)$: if you have the flu, you do not pass the course or if you miss the final examination, then you do not pass the course.

Therefore, English sentence of the proposition role="math" localid="1663698796506" $\left(p\to \neg r\right)\vee \left(q\to \neg r\right)$is

If you have the flu, you do not pass the course or

If you miss the final examination or you pass the course.

(f) (p∧q)∨(¬q∧r)

Consider the following proposition:

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

Express to the given proposition as an English sentence.

Recall that $p\wedge q$is equivalent to $p$and $q$is equivalent to $p$and $q$.

Recall that $p\vee q$is equivalent to $p$or $q$.

$p\wedge q:$you have the flu and you miss the final examination.

:$\neg q$: you do not miss the final examination.

:$q\to \wedge r$ you do not miss the final examination and you pass thecourse.

:$\left(p\wedge q\right)\vee \left(\neg q\wedge r\right)$:you have the flu and you miss the final examination, or you do not miss the final examination and you pass the course.

Therefore, English sentence of the proposition $\left(p\wedge q\right)\vee \left(q\wedge r\right)$is

you have the flu and you miss the final examination,

you do not miss the final examination and you pass the course.