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Chapter 3: Problem 7

Use a truth table to determine whether the two statements are equivalent. \((p \wedge q) \wedge r, p \wedge(q \wedge r)\)

Short Answer

Expert verified
Since conjunction (AND operator), represented by \(\wedge\), is associative, the two statements \((p \wedge q) \wedge r\) and \(p \wedge (q \wedge r)\) are logically equivalent. The truth values from the truth table for both statements would be the same. Therefore, they are equivalent.

Step by step solution

01

Set up a truth table

The truth table should include a column for each variable (p, q, r) and a column for the overall truth value of each statement. The overall truth value is determined by applying the AND operator to the variables as specified by the statement. The table should have 8 rows corresponding to all possible combinations of truth values for the three variables.
02

Fill out the truth table

Fill out the truth table by systematically working through all the possible combinations of truth values for the three variables. Record the truth value of each statement for each combination of variables. The AND operator (denoted by \(\wedge\)) returns true if and only if both of its operands are true.
03

Compare the columns

Compare the two columns containing the overall truth values of the two statements. If the two columns are identical, then the two statements are equivalent. If there are any differences, then the two statements are not equivalent.

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Most popular questions from this chapter

Write an original argument in words for the direct reasoning form.

In this section, we used a variety of examples, including arguments from the Menendez trial, the inevitability of Nixon's impeachment, Spock's (fallacious) logic on Star Trek, and even two cartoons, to illustrate symbolic arguments. a. From any source that is of particular interest to you (these can be the words of someone you truly admire or a person who really gets under your skin), select a paragraph or two in which the writer argues a particular point. (An intriguing source is What Is Your Dangerous Idea?, edited by John Brockman, published by Harper Perennial, 2007.) Rewrite the reasoning in the form of an argument using words. Then translate the argument into symbolic form and use a truth table to determine if it is valid or invalid. b. Each group member should share the selected passage with other people in the group. Explain how it was expressed in argument form. Then tell why the argument is valid or invalid.

Use Euler diagrams to determine whether each argument is valid or invalid. All dancers are athletes. Savion Glover is a dancer. Therefore, Savion Glover is an athlete.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used Euler diagrams to determine that an argument is valid, but when I reverse one of the premises and the conclusion, this new argument is invalid.

Use the standard forms of valid arguments to draw a valid conclusion from the given premises. If I am a full-time student, I cannot work. If I cannot work, I cannot afford a rental apartment costing more than \(\$ 500\) per month. Therefore, ...
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