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Chapter 3: Problem 91

Describe how to construct a truth table for a compound statement.

Short Answer

Expert verified
To construct a truth table for a compound statement 'p and q', first identify the components 'p' and 'q'. Then, create a four-row table covering all possible combinations of truth values for 'p' and 'q'. Finally, calculate the truth value of 'p and q' for each row based on the logical 'AND' operator, resulting in the values 'True', 'False', 'False', 'False' respectively.

Step by step solution

01

Identify Components

Identify the components of the compound statement. A compound statement is made up of two or more simple statements (also called sub-statements). For example, the compound statement 'p and q' has two components: 'p' and 'q'.
02

Create Table

Create a truth table that includes a column for each component of the compound statement. If there are n components, the truth table should have \(2^n\) rows. This is to cover all possible truth value combinations. Each column should have alternate sequence of true (T) and false (F) values, starting with T. With each next column, the period of alternation doubles. For example, if we have two components 'p' and 'q', the truth table starts as: \n\n| p | q |\n|---|---|\n| T | T |\n| T | F |\n| F | T |\n| F | F |
03

Calculate Compound Statement Values

Calculate the truth value of the compound statement for each row of the table. The values should be computed using the logical connectives in the compound statement. For example, with the compound statement 'p and q', the resulting truth table completes as: \n\n| p | q | p AND q |\n|---|---|---|\n| T | T | T |\n| T | F | F |\n| F | T | F |\n| F | F | F |

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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 you tell me what I already understand, you do not enlarge my understanding. If you tell me something that I do not understand, then your remarks are unintelligible to me. \(\therefore\) Whatever you tell me does not enlarge my understanding or is unintelligible to me.

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 it rains or snows, then I read. I am not reading. \(\therefore\) It is neither raining nor snowing.

Use Euler diagrams to determine whether each argument is valid or invalid. All insects have six legs. No spiders are insects. Therefore, no spiders have six legs.

Use a truth table to determine whether the symbolic form of the argument is valid or invalid. $$ \begin{aligned} &q \rightarrow \sim p \\ &q \wedge r \\ &\therefore r \rightarrow p \end{aligned} $$

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.
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