Problem 6 - 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. \(p \wedge \sim q\)

Step-by-Step Solution

Answer

The truth value for \( p \wedge \sim q \) is true.

Step 1: Evaluate the truth value of the statements

Evaluate if 4+6=10, this is true, hence \( p \) is true. Then evaluate if 5*8=80, this is false. Thus, \( q \) is false

Step 2: Evaluate the negation of \( q \)

The statement \( q \) is false. The negation of a false statement is true. Thus, \( ∼ q \) is true

Step 3: Evaluate the conjunction \( p \wedge ∼ q \)

Since the statement \( p \) is true and \( ∼ q \) is also true, the conjunction \( p \wedge ∼ q \) is hence also true, it being only false if at least one of the statements is false.

Key Concepts

Truth Value

In logic, each statement is either true or false; this is known as its truth value. For example, with the expressions provided: \(4 + 6 = 10\) and \(5 \times 8 = 80\), each has a distinct truth value. - The truth value of \(4 + 6 = 10\) is **true**, because the statement is mathematically correct.- However, the truth value of \(5 \times 8 = 80\) is **false**, since multiplying gives 40, not 80. Understanding truth values is crucial for evaluating logical statements by determining whether they accurately represent facts.

Negation

Negation is about flipping the truth value of a statement. If you have a statement \(q\), its negation is represented as \(\sim q\), which means "not \(q\)".- When the original statement \(q\) is false, its negation \(\sim q\) becomes true.- Conversely, if \(q\) is true, then \(\sim q\) would be false.In this context, since \(q\) is false (because \(5 \times 8 eq 80\)), its negation \(\sim q\) is true. Negation helps us to transform and understand statements from the opposite perspective.

Mathematical Reasoning

Mathematical reasoning refers to the process of using logical thinking to draw conclusions based on mathematical principles.- When reasoning through a problem, we first evaluate individual components, such as the truth values of statements.- Next, we apply logical operations like negation or conjunction to combine the information.By using these steps rationally, we can effectively solve complex logical problems. In this exercise, determining the truth values of \(p\) and \(q\), then applying negation to \(q\), showcases the mathematical reasoning process in action.

Conjunction Evaluation

The conjunction operation, denoted by \(\wedge\), combines two statements and returns true only if both individual statements are true.- For a conjunction \(p \wedge q\), it is only true when both \(p\) and \(q\) are true.- If either \(p\) or \(q\) is false, the result is false.In this case, we have \(p \wedge \sim q\). Since both \(p\) (\(4 + 6 = 10\)) is true and \(\sim q\) (negation of false \(5 \times 8 = 80\)) is true, the conjunction \(p \wedge \sim q\) is true. Evaluating conjunctions helps verify the combined truthfulness of multiple statements.