Problem 13 - Thinking Mathematically

Question

Let \(p\) and q represent the following statements: $$ \begin{aligned} &p: 4+6=10 \\ &q: 5 \times 8=80 \end{aligned} $$ Determine the truth value for each statement. \(p \vee \sim p\)

Step-by-Step Solution

Answer

The truth value of the statement \(p \vee \sim p\) is True.

Step 1: Understand the Proposed Statements

The statement \(p\) represents that '4+6=10' and statement \(q\) represents that '5*8=40'. We must determine the truth value of each statement.

Step 2: Determine the Truth Value of p

The calculation of \(p\) as '4+6=10' is a True statement as the summation is correct, hence the truth value of \(p\) is True.

Step 3: Calculate \(\sim p\)

The symbol \(\sim\) means NOT. So \(\sim p\) would mean NOT p. Since we determined that \(p\) is True, \(\sim p\) will be False.

Step 4: Apply the Law of Logic

The operation \(p \vee \sim p\) represents the logical OR operation. This means, if either \(p\) or \(\sim p\) is true then the final statement \(p \vee \sim p\) is true. Since \(p\) is True and \(\sim p\) is False, the final result is True (because True OR False is True).

Key Concepts

Truth Value

In logic, statements are often evaluated based on their **truth value**. A truth value describes whether a proposition is true or false. In propositional logic, every statement or 'proposition' can be assigned a truth value to provide clarity and precision to logical reasoning.
Consider the statement "4 + 6 = 10" given as \( p \). When we evaluate it logically, since 4 plus 6 indeed equals 10, the truth value of \( p \) is **True**. Understanding and identifying the truth value of statements is crucial as it forms the foundation of more complex logical operations.

Logical OR

A logical OR operation is a fundamental concept in propositional logic. It is denoted by the symbol \( \vee \) and represents a disjunction between two statements. The outcome of a logical OR operation is true if at least one of the propositions is true.
For example, for the expression \( p \vee \sim p \), the OR operation is performed between the truth of \( p \) and the negation of \( p \). Since the first statement, \( p \), is true, the overall truth value of \( p \vee \sim p \) will be True. This is because, for logical OR, having one true proposition results in a true outcome.

Negation

**Negation** is a logical operation that inverts the truth value of a given statement. It is denoted by the symbol \( \sim \) or the word 'NOT'.
In logical terms, if a statement \( A \) is true, then \( \sim A \) (NOT A) will be false, and vice versa. In the exercise, the original statement \( p \) is true. Therefore, its negation, \( \sim p \), is false. Negation is important because it allows us to express and evaluate the opposite or contradiction of any statement.

Propositional Logic

**Propositional logic**, or propositional calculus, concerns the manipulation and combination of boolean truth values. It uses logical connectives such as AND, OR, and NOT to form complex expressions from simpler statements.
In the exercise, the statements \( p \) and \( q \) are basic propositions. Through propositional logic, these statements are analyzed to deduce the truth value of the compound statement \( p \vee \sim p \). Propositional logic is essential for forming arguments and verifying their validity, providing the groundwork for more advanced forms of reasoning and decision making.