Problem 3 - 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. \(p \wedge q\)

Step-by-Step Solution

Answer

The truth value of the statement \(p \wedge q\) is false.

Step 1: Evaluate p

Let's first evaluate the truth value of the statement p, which is '4 + 6 = 10'. By simple arithmetic calculation, we can confirm that 4 + 6 indeed equals 10. So, the statement p is true.

Step 2: Evaluate q

Now, let's evaluate the truth value of the statement q, which states that '5 x 8 = 80'. A quick multiplication operation shows that 5 times 8 is actually 40, not 80. Therefore, the statement q is false.

Step 3: Apply the AND operator to the truth-values of p and q

By definition of the logical AND operator, it returns true only when both operands are true. We have determined that p is true and q is false. Therefore, the statement \(p \wedge q\) is false.

Key Concepts

Truth Value

In logic, determining the truth value of a statement means identifying whether it is true or false. Each statement is like a little declarative sentence. For example, when we say, "4 + 6 = 10," we are making a declaration based on arithmetic.
To find the truth value, we must determine if this statement is accurate. In our case above, since 4 plus 6 equals 10, the truth value of "4 + 6 = 10" is true. If a statement doesn't conform to the facts or known information, it is false.

• True: The statement correctly describes a fact (e.g., 4 + 6 = 10).
• False: The statement incorrectly describes a fact (e.g., 5 x 8 = 80 is false because it actually equals 40).

Learning to evaluate truth values is fundamental in logic and mathematics as it lays the groundwork for understanding more complex logical expressions.

Logical AND

The logical AND operator, denoted as \( \wedge \), is crucial in constructing compound logical statements. It combines two simpler statements into a more complex one. For instance, if we have two statements, \(p\) and \(q\), we can form \(p \wedge q\).
For an AND statement to be true, both component statements must individually be true. This truth condition is strict.

• If \(p\) is true and \(q\) is true, then \(p \wedge q\) is true.
• If \(p\) is true and \(q\) is false, \(p \wedge q\) ends up being false.
• Even if only one statement is false, \(p \wedge q\) is false.

Thus, the logical AND requires the simultaneous truth of both parts to hold the entire statement true.

Statement Evaluation

Statement evaluation is the process of assessing whether a given statement is valid by analyzing its components. This involves breaking down the statement into simpler parts, evaluating each one, and then applying logical operations.
When evaluating, follow these steps:

• Analyze each basic statement separately. For example, \(p\): "4 + 6 = 10," which is true, and \(q\): "5 \times 8 = 80," which is false.
• Apply logical operators to these assessments. We combined \(p\) and \(q\) using the logical AND \( \wedge \).
• Determine the final truth value based on the rules of logical AND.

By thoroughly evaluating each part of a compound statement, one can ascertain the overall correctness of more complex logical propositions and develop a robust understanding of logical reasoning.