Problem 10 - 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. \(p \vee q\)

Step-by-Step Solution

Answer

The truth value of the statement 'p OR q' is true.

Step 1: Determine the truth value of statement p

The mathematical expression for statement p is 4+6=10. Check if this statement is true. The sum of 4 and 6 is 10 which matches the given expression. Hence, the statement \( p \) is true.

Step 2: Determine the truth value of statement q

The mathematical expression for statement q is 5 * 8 = 80. In reality, 5 multiplied by 8 equals 40, not 80. Hence, the statement \( q \) is false.

Step 3: Determine the truth value of 'p OR q'

In logic, 'p OR q' is true if at least one of the statements (p or q) is true. From the previous steps, we have determined that p is true and q is false. So, it can be concluded that 'p OR q' is true because at least one statement, \( p \), is true.

Key Concepts

Truth Value

In logic, the truth value of a statement indicates whether that statement is true or false. For any given statement, there are only two possibilities: it can either be true (often represented as "T") or false (represented as "F"). In the exercise, we evaluated the truth value of two statements:

• The statement \( p: 4+6=10 \) was found to be true because the calculation is correct.
• The statement \( q: 5 \times 8=80 \) was found to be false because the actual product of 5 and 8 is 40, not 80.

Understanding truth values is essential in logic as they form the basis for evaluating more complex logical expressions. Always verify the calculations involved in a statement to determine its truth value.

Logical Operators

Logical operators are tools used in logic to combine or modify statements, allowing us to form more complex expressions. The basic logical operators include AND (\( \land \)), OR (\( \lor \)), and NOT (\( eg \)). These operators help us build compound statements from simpler ones.

• AND (\( \land \)) results in true if both statements are true. It combines statements where both conditions must be met.
• OR (\( \lor \)) results in true if at least one of the statements is true. It offers flexibility as only one statement needs to be true for the entire expression to evaluate as true.
• NOT (\( eg \)) negates the truth value of a statement. It flips true to false and vice versa, which is useful for forming statements that require the opposite condition to hold.

In the exercise, we especially focused on the OR logical operator, as it is used to evaluate the statement "\( p \vee q \)." Understanding logical operators is fundamental for constructing and interpreting logical expressions effectively.

Disjunction

The disjunction in logic is represented by the OR operator \( (\lor) \). It is a compound statement that connects two statements such that the compound statement is true if at least one of the component statements is true. This allows for more inclusive conditions in logical reasoning.In the exercise, the disjunction "\( p \vee q \)" was evaluated using the truth values determined from the individual statements:

• \( p \) was true.
• \( q \) was false.

The disjunctive expression \( p \vee q \) is evaluated as true because at least one of the statements, specifically \( p \), holds true. This property of disjunction makes it a versatile tool in scenarios where multiple conditions are evaluated, but not all need to be satisfied for a positive outcome. Embracing disjunction helps in understanding scenarios where possibilities rather than certainties are considered.