Problem 4 - 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. \(q \wedge p\)

Step-by-Step Solution

Answer

The truth value of the statement 'p ∧ q' is false.

Step 1: Determine the Truth Value for 'p'

First, evaluate the statement 'p' which is defined as '4 + 6 = 10'. By doing simple arithmetic, it's clear that '4 + 6' indeed equals '10', so the statement 'p' is true.

Step 2: Determine the Truth Value for 'q'

Second, evaluate the statement 'q' which is defined as '5 × 8 = 80'. Performing the multiplication shows that '5 × 8' equals '40', not '80'. So, the statement 'q' is false.

Step 3: Apply the Logical Operator

The logical operator '∧' (wedge or 'and') is applied to 'p' and 'q'. The statement 'p ∧ q' is true if both 'p' and 'q' are true. Since 'p' is true and 'q' is false, the overall statement 'p ∧ q' is false.

Key Concepts

Understanding Truth Values

In logical reasoning, truth values are fundamental. Truth values denote whether a statement is true or false.

When evaluating statements in logic, we assign a value of 'True' or 'False' to each statement based on whether it accurately describes a fact or not. For example, in the exercise, the statements are:

Step by step, you verify each statement:

• 'p' is True because 4 plus 6 equals 10.
• 'q' is False because 5 times 8 equals 40, not 80.

Truth values help us understand and dissect logical constructs. Recognizing these values allows us to combine statements using logical operators.

Decoding Logical Operators

Logical operators are symbols or words that connect two or more statements to form a new statement.

They are essential tools in logical reasoning and computer science.

For instance, common logical operators include:

• The "and" operator (∧), which results in True only if both combined statements are True.
• The "or" operator (∨), which results in True if at least one of the statements is True.
• The "not" operator (¬), which inverts the truth value.

In the given exercise, 'p ∧ q' uses the 'and' operator. Both statements need to be True for the overall expression to be True.

Since 'p' is True and 'q' is False, applying the "and" operator results in a False value because both conditions aren't met.

Exploring Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division.

They are the basic building blocks of mathematics. These operations follow specific rules and order of operations.

Looking at the statements in the exercise:

• '4 + 6 = 10' involves addition and confirms that 4 plus 6 indeed equals 10.
• '5 × 8 = 80' uses multiplication, but results in 40, revealing the statement as incorrect.

A solid grasp of arithmetic operations can help you evaluate mathematical statements quickly and accurately.

By understanding these operations, you become efficient at solving mathematical problems and verifying the truth of statements.