# Use a truth table to determine whether the argument is valid or invalid. ( p ∨ ∼ r ) → ( q ∧ r ) r ∧ p ∴ p ∨ q

### Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
Publisher: Cengage Learning
ISBN: 9781305965584

Chapter
Section

### Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
Publisher: Cengage Learning
ISBN: 9781305965584
Chapter 3, Problem 82RE
Textbook Problem
1 views

## Use a truth table to determine whether the argument is valid or invalid. ( p ∨ ∼ r ) → ( q ∧ r ) r ∧ p ∴ p ∨ q

To determine

To identify:

Whether the given argument is valid or invalid using the truth table

### Explanation of Solution

Given information:

The argument (p~r)(qr)rp_pq

Concept Involved:

An Argument and a Valid Argument: An argument consists of a set of statements called premises and another statement called the conclusion. An argument is valid if the conclusion is true whenever all the premises are assumed to be true. An argument is invalid if it is not a valid argument.

The following truth table procedure can be used to determine whether an argument is valid or invalid.

Truth Table Procedure to Determine the Validity of an Argument:

1. Write the argument in symbolic form.
2. Construct a truth table that shows the truth value of each premise and the truth value of the conclusion for all combinations of truth values of the simple statements.
3. If the conclusion is true in every row of the truth table in which all the premises are true, the argument is valid. If the conclusion is false in any row in which all the premises are true, the argument is valid.
 The negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the opposite of the truth value of p. A disjunction is a compound statement formed by joining two statements with the connector OR. The disjunction "p or q" is symbolized by p∨  q. A disjunction is false if and only if both statements are false; otherwise it is true. A conjunction is a compound statement formed by joining two statements with the connector AND. The conjunction "p and q" is symbolized by p∧q. A conjunction is true when both of its combined parts are true; otherwise it is false.
 A conditional statement, symbolized by  p→q, is an if-then statement in which p is a hypothesis and q is a conclusion. The logical connector in a conditional statement is denoted by the symbol  →. The conditional is defined to be true unless a true hypothesis leads to a false conclusion. A biconditional statement is defined to be true whenever both parts have the same truth value. The biconditional operator is denoted by a double-headed arrow ↔. The biconditional p↔q represents "p if an only if q", where p is a hypothesis and q is a conclusion.

Calculation:

( p~r )( qr ) rp _ pq | First Premise Second Premise _ Conclusion

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