Skip to main content

Chapter 4 Logical Arguments

Exercises Practice Problems

1. Implications.

Let \(x \in \mathbb{R}\) be a real number.

  1. Write down the contrapositive, converse and inverse of the implication: "if \(x < 3\) then \(x^2 < 9\text{.}\)" Which of these four statements are true? Which ones are false?

  2. Write down the contrapositive, converse and inverse of the implication: "If it is warm and sunny then I go swimming."

2. Equivalent Statements.

  1. Use a truth table to show that \(\neg (P \Rightarrow Q)\) is equivalent to \(P \wedge \neg Q\text{.}\)

    \begin{equation*} \begin{array}{c|c||c|c|c|c} P & Q & P \Rightarrow Q & \neg(P \Rightarrow Q) & \neg Q & P \wedge \neg Q \\ \hline T & T & & & & \\ T & F & & & & \\ F & T & & & &\\ F & F & & & &\\ \end{array} \end{equation*}

  2. Use a truth table to show that \(P \Leftrightarrow Q\) is equivalent to \((P \wedge Q) \vee (\neg P \wedge \neg Q)\text{.}\)

    \begin{equation*} \begin{array}{c|c||c|c|c|c|c} P & Q & P \Leftrightarrow Q & \neg P & \neg Q & P \wedge Q & \neg P \wedge \neg Q & (P \wedge Q) \vee (\neg P \wedge \neg Q)\\ \hline T & T & & & & & & \\ T & F & & & & & & \\ F & T & & & & & & \\ F & F & & & & & & \\ \end{array} \end{equation*}
3. Two Logic Puzzles.

  1. Each of the four cards below has a digit printed on one side and a letter printed on the other side.

    A classmate makes the following claim: "If there is a vowel on one side of a card then there is an odd number on the other side." Which cards must be turned over to check whether this statement is true?

  2. Suppose that four women are drinking in a bar. The owner wants to enforce the rule that no one under twenty-one is allowed to drink alcohol. The owner asks each person to tell him either (1) what she is drinking, or (2) how old they are. Here are their truthful answers:

    Table 4.0.1.
    Alice Barbara Clarice Daria
    "I am drinking Coke." "I am sixteen." "I am drinking beer." "I am twenty-five."

    Which of the people have to answer the additional question (that they haven't answered yet) in order for the owner to ensure that the law is being observed?

  3. Which of the previous questions was easier, (a) or (b)? Why?

4. Evaluating an Argument.

Determine the validity of the following arguments. Include a truth table to justify your conclusion.

  1. If I stay up late at night, then I will be tired in the morning. I am tired this morning. Therefore, I stayed up late last night.

  2. If I stay up late at night, then I will be tired in the morning. I am not tired this morning. Therefore, I did not stay up late last night.

5. Evaluating Another Argument.

Determine the validity of the following argument. Include a truth table to justify your conclusion.

If I work hard, then I earn lots of money. If I earn lots of money, then I pay high taxes. Therefore, if I pay high taxes, then I have worked hard.