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Chapter 3: Problem 8

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

Short Answer

Expert verified
The two logical expressions \((p \vee q) \vee r\) and \(p \vee (q \vee r)\) are equivalent. This is demonstrated through a truth table, where each possible combination of truth values for \(p\), \(q\), and \(r\) result in the same truth value for both expressions, verifying the principle of associativity for the logical OR operator.

Step by step solution

01

Understand the logical operations

Logical OR operation (\( \vee \)) returns true if either one of the operands or both are true. Basic logical operations operate on simple true or false values. Here, \(p\), \(q\), and \(r\) are boolean variables, and the OR operation is subject to the principle of associativity, which is what we're testing.
02

Set up the Truth Table

A truth table for three boolean variables (like \(p\), \(q\), and \(r\), in this case) requires 8 rows representing all possible combinations of true (T) and false (F) for these variables. So set up a truth table with eight rows and five columns. The first three columns are for the variables \(p\), \(q\), and \(r\) respectively. The last two columns are for the two statements, \((p \vee q) \vee r\) and \(p \vee (q \vee r)\). The order of the rows for possible variable values should be TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF.
03

Fill in the Truth Table

Fill in the truth values for the both statements for each of the rows. For each possible combination of \(p\), \(q\), and \(r\), calculate the output of \((p \vee q) \vee r\) and \(p \vee (q \vee r)\). For example, when \(p\), \(q\), and \(r\) are all true, both statements become true (because the OR operation on true values results in true). Repeat this process for all combinations.
04

Compare the results

Now, observe the last two columns of the truth table (representing the results of the two statements). If for every possible combination of \(p\), \(q\), and \(r\), the results of the two statements are equal, then the two statements are equivalent. If there is even one combination for which the results are different, then the statements are not equivalent.

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Most popular questions from this chapter

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If I am at the beach, then I swim in the ocean. If I swim in the ocean, then I feel refreshed. \(\therefore\) If I am at the beach, then I feel refreshed.

Write a valid argument on one of the following questions. If you can, write valid arguments on both sides. a. Should the death penalty be abolished? b. Should Roe v. Wade be overturned? c. Are online classes a good idea? d. Should marijuana be legalized? e. Should grades be abolished? f. Should same-sex marriage be legalized?

Exercises 59-60 illustrate arguments that have appeared in cartoons. Each argument is restated below the cartoon. Translate the argument into symbolic form and then determine whether it is valid or invalid. If you do not know how to read, you cannot read War and Peace. If you cannot read War and Peace, then Leo Tolstoy will hate you. Therefore, if you do not know how to read, Leo Tolstoy will hate you.

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If all people obey the law, then no jails are needed. Some jails are needed. \(\therefore\) Some people do not obey the law.

Determine whether each argument is valid or invalid. Some natural numbers are even, all natural numbers are whole numbers, and all whole numbers are integers. Thus, some integers are even.
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