Problem 7 - 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. \(\sim p \wedge \sim q\)

Step-by-Step Solution

Answer

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

Step 1: Evaluating Statement p

The statement \(p: 4+6=10\) is evaluated. The left-hand side (4+6) equates to 10, which is precisely the right-hand side. As a result, the statement p is true.

Step 2: Evaluating Statement q

The statement \(q: 5 \times 8=80\) is evaluated next. The left-hand side (5x8) equates to 40, which doesn't match the right-hand side (80). As a consequence, the statement q is false.

Step 3: Apply Logical Operators

We apply the logical operators to determine the truth value of \(\sim p \wedge \sim q\) (not p and not q). The 'not' operator reverses the truth value of the statement it precedes, so \(\sim p\) is false (since p is true), and \(\sim q\) is true (since q is false). The 'and' operator only returns true if both components are true. So, \(\sim p \wedge \sim q\) is false (since \(\sim p\) is false and \(\sim q\) is true).

Key Concepts

Truth Values

Truth values are the fundamental building blocks of logical reasoning. In logic, any statement can either be true or false. There are no in-betweens or "maybe" options. This binary nature helps us to clearly categorize statements and their relationships. When evaluating a mathematical statement, the truth value refers to whether the statement is mathematically correct or not. For instance, for the statements given:

• Statement \( p: 4+6=10 \), the truth value is "true" because indeed, 4 plus 6 equals 10.

• Statement \( q: 5 imes 8=80 \), the truth value is "false" because 5 times 8 is 40, not 80.

These truth values serve as a foundation for determining the overall truth of more complex logical expressions.

Logical Operators

Logical operators are symbols or words used to connect two or more truth values to form a new statement. The primary logical operators include 'and', 'or', and 'not'. They help us reason about complex scenarios by combining simpler statements.

• The "not" operator (\( \sim \)) changes the truth value of a statement. For a true statement, applying "not" makes it false, and vice versa.

• The "and" operator (\( \wedge \)) combines two statements and only returns "true" if both statements are true.

In the exercise, combination of these operators with statements like \( (\sim p \wedge \sim q) \) helps determine a final truth value.

Negation

The negation, symbolized by \( \sim \), is an operator used to reverse, or negate, the truth value of a statement. It's quite straightforward:

• If a statement is true, its negation is false.
• If a statement is false, its negation is true.

In our exercise, we apply negation to both statements:

• For \( p \), which is true, \( \sim p \) becomes false.
• For \( q \), which is false, \( \sim q \) becomes true.

Basic logical tasks, like this, are crucial for understanding more advanced logical operations.

Conjunction

Conjunction is represented by the "and" operator (\( \wedge \)), and its primary role is to evaluate the truth of two combined statements. In a conjunction, both statements need to be true for the overall expression to be true. If either statement is false, the conjunction as a whole becomes false.In our problem, we perform a conjunction of the negated statements:

• The statement \( \sim p \) is false.
• The statement \( \sim q \) is true.
• Thus, the conjunction \( \sim p \wedge \sim q \) results in false because "and" demands both components be true, which they are not.

Understanding conjunction helps us build complex logical expressions and determine their truthfulness.