Problem 9 - 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. \(q \vee p\)

Step-by-Step Solution

Answer

The truth value for the statement \(p \vee q\) is true.

Step 1: Identifying The Statements

The first step is to identify the two given statements. We have \(p: 4 + 6 = 10\) and \(q: 5 \times 8 = 80\).

Step 2: Evaluate Statement p

In the statement \(p\), we can see that 4 + 6 = 10. This is a true statement.

Step 3: Evaluate Statement q

In the statement \(q\), we can see that 5 \times 8 = 40. But according to the given statement, 5 \times 8 = 80. So, this is a false statement.

Step 4: Apply Logical OR operation

The OR operator \( \vee \) states that if any of the statements are true, then \(p \vee q\) is true. Since statement p is true, \(p \vee q\) stands true, regardless of the truth value of q.

Key Concepts

Propositional Logic

In logic and mathematics, propositional logic forms the foundation of logical reasoning. It deals with propositions, which are statements that are either true or false. These propositions do not rely on specific values or variables, but on the form of the expressions themselves. When you work with propositional logic, you're essentially evaluating statements to determine their truth values, based on logical expressions.Propositional logic is composed of simple or compound statements. Simple statements are checked independently for their truth conditions, while compound statements consist of simpler ones combined using logical connectors or operators like AND (\( \wedge \)), OR (\( \vee \)), NOT (\( eg \)), etc. By using these operators, propositional logic evaluates the combined statement's truth value based on the truthfulness of its components.

Truth Value Evaluation

Truth value evaluation is a central concept in logic involving determining the truthfulness of a given proposition. For instance, in the exercise, we had two statements, \(p\) and \(q\). To evaluate each, we substitute the numerical expressions to see if they hold true.

• For statement \(p\) ("4 + 6 = 10"), calculating 4 plus 6 gives us 10, hence \(p\) is true.
• For statement \(q\) ("5 \times 8 = 80"), we see that 5 times 8 equals 40, not 80, making \(q\) false.

Through evaluation, we assign either a true or false value to each proposition. Mastery of this concept helps in logically analyzing larger, complex statements by breaking them down into true or false parts.

Logical Operators

Logical operators are crucial in forming compound propositions. These operators dictate how propositions are interconnected and influence the truth value of the overall logical expression.In this exercise, the OR operator (\( \vee \)) is used. The OR operator is unique because it results in a true proposition if at least one of the operands (or propositions) is true. Using \(p \vee q\), as long as \(p\) or \(q\) is true, the entire expression is true.Consider:

• If both \(p\) and \(q\) were true, \(p \vee q\) would be true.
• If \(p\) is true and \(q\) is false, while \(p \vee q\) remains true.
• If both \(p\) and \(q\) are false, then \(p \vee q\) is false.

Understanding how these operators work is vital to constructing and deconstructing logical arguments and allows for a clearer interpretation of their outcomes.