Problem 59 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The truth value of the given statement when \(p\) is false, \(q\) is true, and \(r\) is false is false.

Step 1: Replace the variables with their truth values

First, substitute the given truth values into the statement. Thus, the statement becomes \(\sim(\text{false} \vee \text{true}) \wedge \sim(\text{false} \wedge \text{false})\).

Step 2: Evaluate the expressions inside parentheses

To evaluate each part of the expression, know that 'OR' returns true if at least one of the inputs is true, while 'AND' returns true if both inputs are true. Therefore, \(\text{false} \vee \text{true}\) is true and \(\text{false} \wedge \text{false}\) is false. The statement now becomes \(\sim(\text{true}) \wedge \sim(\text{false})\).

Step 3: Apply the NOT operator

The 'NOT' operator inverts the truth value of the expression it is applied to. Thus, \(\sim(\text{true})\) becomes false and \(\sim(\text{false})\) becomes true. The statement is now \(\text{false} \wedge \text{true}\).

Step 4: Evaluate the remaining AND operation

As mentioned earlier, the 'AND' operator produces true only if both inputs are true. Hence, \(\text{false} \wedge \text{true}\) yields false. This gives the final result.

Key Concepts

Truth Value

In propositional logic, each statement or proposition can have a truth value of either true or false. These are the fundamental "building blocks" of logical expressions.
Whenever you evaluate a logical statement, your goal is to determine whether the statement is true or false based on the given propositions.

For example:- The truth value of a proposition like \(p\) could be false.- Meanwhile, another proposition like \(q\) could be true.
Understanding the truth value of individual propositions is the first step in evaluating more complex logical expressions.

Logical Operators

Logical operators are symbols or words used to connect propositions in propositional logic to create complex statements. The commonly used logical operators are 'AND' (\( \wedge \)), 'OR' (\( \vee \)), and 'NOT' (\( \sim \)). Each operator has a specific way of affecting truth values.

• AND (\( \wedge \)): This operator returns true only when both connected propositions are true. For example, \(p \wedge q\) is true if both \(p\) and \(q\) are true.
• OR (\( \vee \)): This operator yields true if at least one of the propositions is true. That means \(p \vee q\) will be true if either \(p\) is true, \(q\) is true, or both are true.
• NOT (\( \sim \)): This unary operator inverts the truth value of a single proposition. So, \( \sim p\) is true when \(p\) is false, and vice versa.

By mastering these tools, you can evaluate any complex logical expression and determine its overall truth value.

Propositional Logic

Propositional logic is a branch of logic dealing with propositions which are either true or false. It involves using propositions connected by logical operators to form statements that can be analyzed for their truth value.

A proposition is a statement that declares something that can be independently true or false. For example, "it is raining" is a proposition.

A logical expression like \(\sim(p \vee q) \wedge \sim(p \wedge r)\) in propositional logic is built from basic propositions (\(p\), \(q\), \(r\)) and operators (\(\vee\), \(\wedge\), \(\sim\)).
Each proposition gets a truth value, and the expression as a whole is evaluated through logical operations.
The systematic approach to evaluate such expressions includes:

• Substituting truth values of propositions in place of variables.
• Applying logical operators step-by-step, based on precedence and associativity.
• Refining the expression until a single truth value result remains for the entire statement.

This enables clear determination of whether complex logical relations hold true.