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Exercises 3.6 Chapter Test

1.

Write each sentence in symbolic form:

  1. Today is Monday (m) and it is my birthday (b).

  2. If x is divisible by 2 (d), then x is an even number (e).

Answer.

  1. \(\displaystyle m \wedge b\)

  2. \(\displaystyle d \to e\)

2.

Let \(r\) represent "the Rockies will win their next game", and \(w\) represent "I will wear my lucky shirt". Write each statement as a sentence in English.

  1. \(\displaystyle w \to r\)

  2. \(\displaystyle \sim \left( r \wedge w \right)\)

Answer.

  1. If I wear my lucky shirt, the Rockies will win their next game.

  2. It is not true that the Rockies will win their next game and I will wear my lucky shirt.

3.

Negate each statement:

  1. All snakes are poisonous.

  2. Some auto mechanics are incompetent.

  3. If Josh works overtime, he will get next Friday off.

Answer.

  1. Some snakes are not poisonous.

  2. No auto mechanics are incompetent.

  3. Josh works overtime and he doesn't get next Friday off.

4.

Use DeMorgan's Law to write a statement that equivalent to: "It is not true that Carol has 2 years of experience and a bachelor's degree."

Answer.

Carol doesn't have 2 years of experience or she doesn't have a bachelor's degree.

5.

Construct a truth table for

  1. \(\displaystyle \sim \left( p \wedge \sim q \right)\)

  2. \(\displaystyle \left( \sim p \to q \right) \vee \left( \sim q \wedge p \right)\)

  3. \(\displaystyle \sim p \to \left(q \vee p \right)\)

Answer.

  1. TFTT

  2. TTTF

  3. TTTF

Solution.
  1. p q \(\sim\) (\(p\) \(\wedge\) \(\sim q\))
    T T T T F F
    T F F T T T
    F T T F F F
    F F T F F T
  2. p q (\(\sim p\) \(\to\) \(q)\) \(\vee\) (\(\sim q\) \(\wedge\) \(p\))
    T T F T T T F F T
    T F F T F T T T T
    F T T T T T F F F
    F F T F F F T F F
  3. p q \(\sim p\) \(\to\) (\(q\) \(\vee\) \(p\))
    T T F T T T T
    T F F T F T T
    F T T T T T F
    F F T F F F F
6.

Use truth tables to determine if the statements are equivalent:

\(\sim \left( p \wedge \sim q \right)\)

\(\sim p \vee q\)

Answer.
The statements are logically equivalent because they have the same values in truth tables: TFTT
Solution.

p q \(\sim\) (\(p\) \(\wedge\) \(\sim q\)) \(\sim p\) \(\vee\) \(q\)
T T T T F F F T T
T F F T T T F F F
F T T F F F T T T
F F T F F T T T F

7.

Write the converse, inverse, and contrapositive of the statement, "If the car is red, then Alison doesn’t like it."

Answer.

Converse: If Alison doesn't like the car, it is red.

Inverse: If the car is not red, then Alison doesn't like it.

Contrapositive: If Alison likes the car, then it is not red.

8.

Which of the following statements are equivalent?

  1. If it rains, then Erik will go to the movies.

  2. If it doesn’t rain, then Erik won’t go to the movies.

  3. If Erik goes to the movies, then it rained.

  4. If Erik doesn’t go to the movies, then it didn’t rain.

Answer.
Statements a and d are equivalent to each other. Statements b and c are equivalent to each other.
Solution.

We can write the statements in symbolic form as:

  1. \(\displaystyle r \to m\)

  2. \(\displaystyle \sim r \to \sim m\)

  3. \(\displaystyle m \to r\)

  4. \(\displaystyle \sim m \to \sim r\)

Since the conditional and contrapositive are logically equivalent, statements a and d are equivalent. The converse and the inverse are equivalent to each other, so b and c are logically equivalent.

Exercise Group.

Identify the form of each argument. Then determine if the argument is valid or invalid.

9.

If you take a vacation, you will be happy. You take a vacation. Therefore, you are happy.

Answer.

Valid; Law of detachment

10.

If Francis enjoys spicy food, then she will like this curry. She likes this curry. Therefore, she must enjoy spicy food.

Answer.

Invalid; Fallacy of the converse

Exercise Group.
Write each argument in symbolic form. Then use truth tables to determine whether the argument is valid or invalid.
11.

If I eat this cookie, I will not have cake. I didn't eat this cookie. Therefore, I had cake.

Answer.

Invalid argument

Solution.

Let \(c\text{:}\) I eat this cookie, and \(k\text{:}\) I will have cake. The argument in symbolic form is:

\begin{equation*} c \to \sim k \end{equation*}
\begin{equation*} \sim c \end{equation*}
\begin{equation*} \therefore k \end{equation*}

Truth tables for the premises and conclusions are:

Variable: c Variable: k Premise: \(c \to \sim k\) Premise: \(\sim c\) Conclusion: \(k\)
T T F F T
T F T F F
F T T T T
F F T T F

Both premises are true in the last two rows, but the conclusion is false in the last row. This means the argument is invalid.

12.

If David rents a car, then he will visit his sister. David didn't visit his sister. Therefore, he didn't rent a car.

Answer.

Valid argument

Solution.

Let \(r\text{:}\) David rents a car, and \(v\text{:}\) David visits his sister. The argument in symbolic form is:

\begin{equation*} r \to v \end{equation*}
\begin{equation*} \sim v \end{equation*}
\begin{equation*} \therefore \sim r \end{equation*}

Truth tables for the premises and conclusions are:

Variable: r Variable: v Premise: \(r \to v\) Premise: \(\sim v\) Conclusion: \(\sim r\)
T T T F F
T F F T F
F T T F T
F F T T T

Both premises are true in the last row only. Since the conclusion is also true in the last row, the argument is invalid.

13.

Determine whether each statement is an example of inductive or deductive reasoning.

  1. Omar is using the rules of algebra to find the solution to an equation on a test.

  2. When it snowed last week, campus was closed. If it snows again, campus will close.

Answer.

  1. Deductive

  2. Inductive

14.

Identify the logical fallacy in each argument.

  1. You have to give to me raise. Otherwise, I won't be able to pay my bills.

  2. The Denver Broncos are the best football team beacuse there is no team better than the Broncos.

  3. We can go to out to dinner or eat cereal at home. The restaurant is too expensive, so we must eat cereal for dinner.

Answer.

  1. Appeal to consequence

  2. Circular reasoning

  3. False dilemma