Explanation Given: The given argument is, If George prepares gourmet food, then he is a good cook. If Brenda does not prepare gourmet food, then Sarah buys fast food. Sarah does not buy fast food. Therefore, George does not prepare gourmet food, and Brenda does not prepare gourmet food. Approach: The meaning of the logical connectives symbols are as follows, Name Logical Connective Symbol Conjunction and ∧ Disjunction or ∨ Negation not ∼ Conditional If…then → Bi-conditional If and only if ↔ Table ( 1 ) 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 true 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 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 . An argument is valid if and only if the conditional [ P 1 ∧ P 2 ∧ P 3 ∧ ... ∧ P n ] → C is a tautology. Calculation: Consider the statements as follows: p : George prepare gourmet food. q : George is a good cook. r : Brenda does not prepare gourmet food. s : Sarah buys fast food. The symbolic form of the argument is, p → q r → s ∼ s _ ∴ ∼ p ∧ r Therefore, the conditional for the argument is, [ ( p → q ) ∧ ( r → s ) ∧ ∼ s ] → ( ∼ p ∧ r ) There are 4 letters in the given symbolic expression. So, there will be 2 4 = 16 rows in the truth table. One column for p , second column for q , third column for r , fourth column for s , fifth column for ~ p , sixth column for ∼ s , seventh column for p → q , eighth column for r → s , ninth column for ∼ p ∧ r , tenth column for [ ( p → q ) ∧ ( r → s ) ∧ ~ s ] , and eleventh column for the conditional statement [ ( p → q ) ∧ ( r → s ) ∧ ∼ s ] → ( ∼ p ∧ r )

11th Edition

ISBN: 8220100478185

Author: Tan

Publisher: CENGAGE L

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Logic and Proofs

Discrete mathematics deals with the separated values of data such as integers. It is widely used in the practical field of computer science and mathematics. By knowing discrete mathematics one can improve their reasoning and problem-solving capabilities.…

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Chapter A.5, Problem 23E

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The given argument is valid.

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Chapter A.5, Problem 22E

Chapter A.5, Problem 24E

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The given argument is valid.

Solution Summary: The author explains the meaning of the logical connectives symbols, which are as follows: Name Logical Connective Symbol Conjunction and wedge

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The given argument is valid.