Problem 58 - Thinking Mathematically

Question

Determine the truth value for each statement when \(p\) is false, \(q\) is true, and \(r\) is false. \(\sim(p \vee q) \wedge r\)

Step-by-Step Solution

Answer

The truth value of the statement \(\sim(p \vee q) \wedge r\) is false.

Step 1: Interpret Variables

According to the given, \(p\) is false, \(q\) is true and \(r\) is false.

Step 2: Apply Disjunction Operator

Disjunction (\(\vee\)) is the logical OR. It's true if at least one of the operands is true. \(p \vee q\) would hence be false OR true which results in true.

Step 3: Apply Negation Operator

Negation (\(\sim\)) is the logical NOT. It's true if the operand is false, and vice versa. Considering \(\sim(p \vee q)\), this would translate to NOT true, which results in false.

Step 4: Apply Conjunction Operator

The whole expression is now false AND false (because \(r\) is false). Conjunction (\(\wedge\)) is the logical AND. It's true if both operands are true. So false AND false results in false.

Key Concepts

Disjunction Operator

The disjunction operator is one of the fundamental concepts in logical reasoning. It's often represented by the symbol \( \vee \) and is also known as the logical OR. This operator works by evaluating two statements and returning true if at least one of the statements is true.
In the example from the exercise, we have the expression \( p \vee q \). Here, \( p \) stands for a false statement and \( q \) a true one. Since the disjunction requires only one true statement to yield a true result, \( p \vee q \) equates to true.

• If both statements are false, the result is false.
• If either statement is true, or both are true, the result is true.

Understanding this concept is crucial for determining the truth values in logical expressions.

Negation Operator

The negation operator is another key tool in logical reasoning, represented by the symbol \( \sim \) and sometimes simply noted as "NOT". It changes the truth value of a statement: true becomes false, and false becomes true.
In the given exercise, the expression \( \sim(p \vee q) \) employs a negation operator. Here, we've already determined that \( p \vee q \) results in true. Applying negation flips this value, turning the expression into false.

• Negating a true statement yields false.
• Negating a false statement yields true.

Negation is used when it's needed to assert that a particular condition does not hold.

Conjunction Operator

The conjunction operator, often seen as \( \wedge \), represents the logical AND. It evaluates two statements and yields true only if both statements are true.
In our exercise, the expression \( \sim(p \vee q) \wedge r \) involves using the conjunction operator. We've already established that \( \sim(p \vee q) \) is false and \( r \) is also false. When applying conjunction, false AND false results in false.

• Both statements must be true for the conjunction to return true.
• If one or both statements are false, the result is false.

Recognizing how conjunction functions within logical statements is vital to accurately assess the truth value of more complex expressions.