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Chapter 1: Problem 1

Give the three truth tables that define the logical operators \(\wedge, \vee\), and \(\neg\).

Short Answer

Expert verified
Truth tables define \(\wedge\), \(\vee\), and \(\neg\) based on combinations of truth values.

Step by step solution

01

- Truth Table for Conjunction (AND)

The logical operator \(\wedge\) represents an AND operation. To create its truth table, we examine the four possible combinations of truth values for two variables, \(P\) and \(Q\):- When both \(P\) and \(Q\) are true (T), \(P \wedge Q\) is true (T).- When \(P\) is true (T) and \(Q\) is false (F), \(P \wedge Q\) is false (F).- When \(P\) is false (F) and \(Q\) is true (T), \(P \wedge Q\) is false (F).- When both \(P\) and \(Q\) are false (F), \(P \wedge Q\) is false (F).Thus, the truth table for \(\wedge\) is as follows:\[\begin{array}{cc|c}P & Q & P \wedge Q \\hlineT & T & T \T & F & F \F & T & F \F & F & F \\end{array}\]
02

- Truth Table for Disjunction (OR)

The logical operator \(\vee\) represents an OR operation. To derive its truth table, we examine the same combinations of \(P\) and \(Q\):- When both \(P\) and \(Q\) are true (T), \(P \vee Q\) is true (T).- When \(P\) is true (T) and \(Q\) is false (F), \(P \vee Q\) is true (T).- When \(P\) is false (F) and \(Q\) is true (T), \(P \vee Q\) is true (T).- When both \(P\) and \(Q\) are false (F), \(P \vee Q\) is false (F).So, the truth table for \(\vee\) is:\[\begin{array}{cc|c}P & Q & P \vee Q \\hlineT & T & T \T & F & T \F & T & T \F & F & F \\end{array}\]
03

- Truth Table for Negation (NOT)

The logical operator \(eg\) represents the NOT operation and is applied to a single variable. It negates the truth value:- If \(P\) is true (T), \(eg P\) is false (F).- If \(P\) is false (F), \(eg P\) is true (T).Therefore, the truth table for \(eg\) is:\[\begin{array}{c|c}P & eg P \\hlineT & F \F & T \\end{array}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logical Operators
Logical operators are essential in understanding how logical statements can be constructed and evaluated. They are widely used in mathematics, computer science, and logic. The primary logical operators include 'AND', 'OR', and 'NOT'. These operators help in combining or modifying statements to form complex logical expressions.

  • Conjunction (AND) is denoted by the symbol \( \wedge \). This logical operation checks if both statements are true.
  • Disjunction (OR) is represented by the symbol \( \vee \). It evaluates to true if at least one of the statements is true.
  • Negation (NOT), symbolized by \( eg \), is used to invert the truth value of a single statement.

Logical operators form the foundation of truth tables, where different combinations of truth values are recorded to determine the outcome of logical expressions.
Conjunction (AND)
The conjunction operation uses the logical operator \( \wedge \), connecting two propositions to form a single statement. Its truth table demonstrates four combinations of truth values for propositions \(P\) and \(Q\).

  • If both \(P\) and \(Q\) are true, the conjunction \(P \wedge Q\) results in true. This is the only scenario where the conjunction holds true.
  • If \(P\) is true and \(Q\) is false, \(P \wedge Q\) is false. This shows that both conditions must be met for a true outcome.
  • Similarly, if \(P\) is false and \(Q\) is true, the conjunction \(P \wedge Q\) becomes false.
  • Finally, if both \(P\) and \(Q\) are false, \(P \wedge Q\) is false too.

A conjunction is stringent in requirement: both components must be true for the resultant statement to be true. This behavior makes it useful in scenarios needing strict conditions.
Disjunction (OR)
Disjunction uses the symbol \( \vee \) to join two propositions, creating a compound statement. The disjunction expresses a more lenient logic compared to conjunction, allowing for greater flexibility in truth conditions.

  • For propositions \(P\) and \(Q\), if both are true, \(P \vee Q\) remains true, reflecting the inclusivity of OR.
  • If \(P\) is true while \(Q\) is false, \(P \vee Q\) still yields a true result. Here, only one true component suffices.
  • When \(P\) is false but \(Q\) is true, the outcome \(P \vee Q\) also holds true.
  • It is only when both \(P\) and \(Q\) are false that \(P \vee Q\) results in a false statement.

The disjunction operation plays a vital role in situations where flexibility is required and any true condition fulfills a requirement. It is widely used in decision-making processes, programming conditions, and more.

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Most popular questions from this chapter

The sentence "Someone has the answer to every question" is ambiguous. Give two translations of this sentence into predicate logic, and explain the difference in meaning.

Give the converse and the contrapositive of each of the following English sentences: a) If you are good, Santa brings you toys. b) If the package weighs more than one ounce, then you need extra postage. c) If I have a choice, I don't eat eggplant.

Show that \(p \wedge(q \vee r \vee s) \equiv(p \wedge q) \vee(p \wedge r) \vee(p \wedge s)\) for any propositions \(p, q\) \(r\), and \(s .\) In words, we can say that conjunction distributes over a disjunction of three terms. (Recall that the \(\wedge\) operator is called conjunction and \(\vee\) is called disjunction.) Translate into logic and verify the fact that conjunction distributes over a disjunction of four terms. Argue that, in fact, conjunction distributes over a disjunction of any number of terms.

Insert parentheses into the following compound propositions to show the order in which the operators are evaluated: a) \(\neg p \vee q\) b) \(p \wedge q \vee \neg p\) c) \(p \vee q \wedge r\) d) \(p \wedge \neg q \vee r\)

Suppose that the domain of discourse for a predicate \(P\) contains only two entities. Show that \(\forall x P(x)\) is equivalent to a conjunction of two simple propositions, and \(\exists x P(x)\) is equivalent to a disjunction. Show that in this case, DeMorgan's Laws for propositional logic and DeMorgan's Laws for predicate logic actually say exactly the same thing. Extend the results to a domain of discourse that contains exactly three entities.
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