Exercises 2.3 Exercises
1.
Consider the sentence “Bob is wearing flip flops or it is cold out.”
Choosing your own letters, symbolize this statement.
Write a truth table for this statement.
In the northeastern United States there is an unspoken but universally acknowledged competition to be the last man in shorts and flip-flops.
2.
Let \(p\) stand for “I attended class” and \(q\) stand for “I did the homework.”
Symbolize “I attended class but I did not do the homework.”
Write a truth table for this statement.
In English, the word “but” means that two statements are simultaneously, maybe surprisingly, both true.
The statement is \(p \wedge \neg q\text{.}\)
| \(p\) | \(q\) | \(\neg q\) | \(p \wedge \neg q\) |
| \(T\) | \(T\) | \(F\) | \(T\) |
| \(T\) | \(F\) | \(T\) | \(T\) |
| \(F\) | \(T\) | \(F\) | \(F\) |
| \(F\) | \(F\) | \(T\) | \(F\) |
3.
Should the statement “Alicia is doing homework or playing video games” be symbolized as \(p \vee q\) or \(p \oplus q\text{?}\) Explain your reasoning.
Hint.Can Alicia do homework and play video games at the same time?
4.
Let \(p\) be the statement “The teachers unionize”, let \(q\) be the statement “The mayor listens to their demands”, \(r\) be “The students can go to school”, and \(s\) be “The teachers can work safely”. Symbolize “If the teachers unionize, then the mayor listens to their demands or the students can't go to school.”
Solution.5.
If \(r\) is the statement “I attended class”, \(s\) is the statement “I did the homework”, and \(t\) is the statement “I passed the exam”, symbolize (write with symbols) the statement “If I attend class but don't do the homework, then I will fail the exam.”
6.
If \(p\) is the statement “The tacos have cilantro”, \(q\) is the statement “Ash will have the tacos”, and \(r\) is the statement “Ash will have a cola”, symbolize the statement “Ash will have the tacos and a cola, if and only if the tacos don't have cilantro.”
7.
Let \(p\) be the statement “The teachers unionize”, let \(q\) be the statement “The mayor listens to their demands”, \(r\) be “The students can go to school”, and \(s\) be “The teachers can work safely”. Write the statement \((q \wedge s) \leftrightarrow p\) in English.
Solution.Solution: The mayor will listen to their demands and the teachers can work safely if and only if the teachers unionize.
8.
If \(r\) is the statement “I attended class”, \(s\) is the statement “I did the homework”, and \(t\) is the statement “I passed the exam”, write the statement \(t \leftrightarrow (\neg r \vee \neg s)\) in English.
9.
If \(p\) is the statement “The tacos have cilantro”, \(q\) is the statement “Ash will have the tacos”, and \(r\) is the statement “Ash will have a cola”, write the statement \(r \vee (p \to q)\) in English.
10.
Prove \(p \leftrightarrow q\) and \(q \leftrightarrow p\) are equivalent using a method of your choosing.
11.
In this exercise you will explore the relationship between a conditional statement \(p \to q\text{,}\) its converse \(q \to p\text{,}\) its inverse \(\neg p \to \neg q\text{,}\) and its contrapositive \(\neg q \to \neg p\text{.}\)
Prove using a truth table that a conditional is equivalent to its contrapositive.
Prove using a truth table that a conditional is not equivalent to its inverse.
There is one other equivalence among these four statements. What is it? (Find it, but you don't need to prove it.)
12.
Suppose that it whenever it rains, Alicia brings her umbrella. Answer the following questions and in both cases explain your reasoning.
If Alicia brings her umbrella, can you can conclude whether it is raining?
If Alicia does not bring her umbrella, can you conclude whether it is raining?
13.
Suppose the following sentence is true: “If an integer is odd and greater than \(2\text{,}\) then it is prime.” (This sentence is false, but that doesn't matter—suppose it is true.)
Suppose further that we have a integer, \(n\text{.}\) Can we conclude that \(n\) is odd and greater than \(2\) if...
\(n\) is prime?
\(n\) is not prime?
In both answers, explain your reasoning.
14.
Using one or more truth tables, determine which pairs of the following statements are and are not equivalent.
\(\displaystyle p \leftrightarrow q\)
\(\displaystyle \neg p \leftrightarrow q\)
\(\displaystyle p \leftrightarrow \neg q\)
\(\displaystyle \neg p \leftrightarrow \neg q\)
15.
Prove that \(\neg(q \to r) \wedge s\) and \(q \wedge (s \to r)\) are not equivalent using a truth table.
16.
Prove modus ponens using a truth table.
17.
Write down a truth table for the statement \((p \wedge r) \to (q \vee \neg s)\text{.}\)
18.
Write a truth table for the statement \(s \leftrightarrow \neg(t \wedge u \wedge \neg v)\text{.}\)
19.
Show that
is a rule of inference by using a truth table.
This generalizes to the technique proof by cases, which proves
by proving \(p_1 \to q\text{,}\) \(p_2 \to q\text{,}\) etc., \(p_n \to q\text{.}\)
Hint.Hint: It is enough to show that there is no row where \(p \to s\text{,}\) \(q \to s\text{,}\) and \(r \to s\) are all true while \((p \vee q \vee r) \to s\) is false.
20.
Prove that
is a tautology by using a truth table.
It is common in mathematical writing to claim that three statements are equivalent. This claim can be proven by showing that the statements imply each other “in a circle,” like in the above rule of inference. Verify for yourself that the above statement also implies that \(p\) and \(q\) are equivalent, as are \(q\) and \(r\text{.}\)
Hint.Hint: It is enough to show that there is no row where \(p \to q\text{,}\) \(q \to r\text{,}\) and \(r \to p\) are all true while \((p \leftrightarrow r)\) is false.