Explanation Given: The given argument is, 1. Not being eligible to vote is sufficient for ignoring politics. 2. Not being a convicted felon is necessary for being eligible to vote. 3. Ignoring politics is sufficient for being naive. Therefore, being naive is necessary being a convicted felon. Definition: The disjunction is true if at least one of the components of the disjunction is true. The disjunction is false if both the statements are false. The conjunction is true if both the components of the conjunction is true. The conjunction is false if at least one of the statements is false. If the statement p is true the conditional statement will depend on whether the statement q is true or false. If the statement p is false the conditional statement will be false and does not depend on whether the statement q is true or false. The negation ∼ p is the opposite of the statement p . If the component of the statement is true the negation of statement is false and vice-versa. An argument is valid if the conclusion of the argument is guaranteed under its given set of premises. An argument having n premises P 1 , P 2 , P 3 , ... , P n and conclusion C can be represented by the conditional [ P 1 ∧ P 2 ∧ P 3 ∧ ... ∧ P n ] → C . Approach: The given argument is, 1. Not being eligible to vote is sufficient for ignoring politics. 2. Not being a convicted felon is necessary for being eligible to vote. 3. Ignoring politics is sufficient for being naive. Therefore, being naive is necessary being a convicted felon. Let, p : You are eligible to vote, q : You ignoring politics, r : You are a convicted felon, and s : You are a naive. Now, the given argument can be rephrased as, 1. If you are not eligible to vote, then you can ignore politics. 2. If you are not a convicted felon, then you are eligible to vote. 3. If you are ignoring politics, then you are a naive. Therefore, if you are a naive, then you are a convicted felon. The above obtained argument in symbolic form is written as, 1. ~ p → q 2. ~ r → p 3. q → s ∴ s → r Therefore the conditional for the argument is, [ ( ~ p → q ) ∧ ( ~ r → p ) ∧ ( q → s ) ] → ( s → r ) There are 3 letters in the given symbolic expression. So, there will be 2 4 = 16 rows in the truth table. There will be 12 columns in the truth table. The truth table for the given statements is, 1 2 3 4 5 p q r s ~ p ~ r ~ p → q ~ r → p q → s 1 ∧ 2 ∧ 3 s → r 4 → 5 T T T T F F T T T T T T T T T F F F T

MATHEMATICS A PRACTICAL ODYSSEY W/ACCESS

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ISBN: 9780357537343

Author: Johnson

Publisher: CENGAGE L

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Chapter 1.5, Problem 34E

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The validity of the given argument by truth table.

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Chapter 1.5, Problem 33E

Chapter 1.5, Problem 35E

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To check: The validity of the given argument by truth table.

Solution Summary: The author evaluates the validity of the given argument by truth table.