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:
- Write the conditional statement \(p \rightarrow q\) in words.
- Write the converse statement \(q \rightarrow p\) in words.
- Write the inverse statement \(\sim p \rightarrow \sim q\) in words.
- 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.
