2.5.0: Exercises

For the following exercises, determine whether the pair of compound statements are logically equivalent by constructing a truth table.

Exercise $\PageIndex{1}$

Converse: $q \rightarrow p$ and inverse: $\sim p \rightarrow \sim q$

Exercise $\PageIndex{2}$

Conditional: $p \rightarrow q$ and contrapositive: $\sim q \rightarrow \sim p$

Exercise $\PageIndex{3}$

Inverse: $\sim p \rightarrow \sim q$ and contrapositive: $\sim q \rightarrow \sim p$

Exercise $\PageIndex{4}$

Conditional: $p \rightarrow q$ and converse: $q \rightarrow p$

Exercise $\PageIndex{5}$

$\sim p \rightarrow q$ and $p \vee \sim q$

Exercise $\PageIndex{6}$

$\sim p \rightarrow q$ and $p \vee q$

Exercise $\PageIndex{7}$

$\sim(p \wedge q)$ and $\sim p \wedge \sim q$

Exercise $\PageIndex{8}$

$\sim(p \wedge q)$ and $\sim p \vee \sim q$

Exercise $\PageIndex{9}$

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

Exercise $\PageIndex{10}$

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

For the following exercises, answer the following:

1. Write the conditional statement $p \rightarrow q$ in words.
2. Write the converse statement $q \rightarrow p$ in words.
3. Write the inverse statement $\sim p \rightarrow \sim q$ in words.
4. Write the contrapositive statement $\sim q \rightarrow \sim p$ in words.

Exercise $\PageIndex{11}$

$p$ : Six is afraid of Seven and $q$ : Seven ate Nine.

Exercise $\PageIndex{12}$

$p$ : Hope is eternal and $q$ : Despair is temporary.

Exercise $\PageIndex{13}$

$p$ : Tom Brady is a quarterback and $q$ : Tom Brady does not play soccer.

Exercise $\PageIndex{14}$

$p$ : Shakira does not sing opera and $q$ : Shakira sings popular music.

Exercise $\PageIndex{15}$

$p$ :The shape does not have three sides and $q$ : The shape is not a triangle.

Exercise $\PageIndex{16}$

$p$ : All birds can fly and $q$ : Emus can fly.

Exercise $\PageIndex{17}$

$p$ : Penguins cannot fly and $q$ : Some birds can fly.

Exercise $\PageIndex{18}$

$p$ : Some superheroes do not wear capes and $q$ : Spiderman is a superhero.

Exercise $\PageIndex{19}$

$p$ : No Pokémon are little ponies and $q$ : Bulbasaur is a Pokémon.

Exercise $\PageIndex{20}$

$p$ : Roses are red, and violets are blue and $q$ : Sugar is sweet, and you are sweet too.

For the following exercises, use the conditional statement: "If Clark Kent is Superman, then Lois Lane is not a reporter," to answer the following questions.

Exercise $\PageIndex{21}$

Write the hypothesis of the conditional statement, label it with a $p$, and determine its truth value.

Exercise $\PageIndex{22}$

Write the conclusion of the conditional statement, label it with a $q$, and determine its truth value.

Exercise $\PageIndex{23}$

Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: "If Clark Kent is not Superman, then Lois Lane is a reporter."

Exercise $\PageIndex{24}$

Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: "If Lois Lane is a reporter, then Clark Kent is not Superman."

Exercise $\PageIndex{25}$

Which form of the conditional is logically equivalent to the converse?

For the following exercises, use the conditional statement: "If The Masked Singer is not a music competition, then Donnie Wahlberg was a member of New Kids on the Block," to answer the following questions.

Exercise $\PageIndex{26}$

Write the hypothesis of the conditional statement, label it with a $p$, and determine its truth value.

Exercise $\PageIndex{27}$

Write the conclusion of the conditional statement, label it with a $q$, and determine its truth value.

Exercise $\PageIndex{28}$

Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: "If Donnie Wahlberg was a member of New Kids on the Block, then The Masked Singer is not a music competition."

Exercise $\PageIndex{29}$

Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: "If The Masked Singer is a music competition, then Donnie Wahlberg was not a member of New Kids on the Block."

Exercise $\PageIndex{30}$

Which form of the conditional is logically equivalent to the contrapositive, $\sim q \rightarrow \sim p$ ?

For the following exercises, use the conditional statement: "If all whales are mammals, then no fish are whales," to answer the following questions.

Exercise $\PageIndex{31}$

Write the hypothesis of the conditional statement, label it with a $p$, and determine its truth value.

Exercise $\PageIndex{32}$

Write the conclusion of the conditional statement, label it with a $q$, and determine its truth value.

Exercise $\PageIndex{33}$

Identify the following statement as the converse, inverse, or contrapositive, and determine its truth value: "If some fish are whales, then some whales are not mammals."

Exercise $\PageIndex{34}$

Write the inverse in words and determine its truth value.

Exercise $\PageIndex{35}$

Write the converse in words and determine its truth value.

For the following exercises, use the conditional statement: "If some parallelograms are rectangles, then some circles are not symmetrical," to answer the following questions.

Exercise $\PageIndex{36}$

Write the hypothesis of the conditional statement, label it with a $p$, and determine its truth value.

Exercise $\PageIndex{37}$

Write the conclusion of the conditional statement, label it with a $q$, and determine its truth value.

Exercise $\PageIndex{38}$

Write the converse in words and determine its truth value.

Exercise $\PageIndex{39}$

Write the contrapositive in words and determine its truth value.

Exercise $\PageIndex{40}$

Write the inverse in words and determine its truth value.