Problem 16 - 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. \(\sim q \vee \sim p\)

Step-by-Step Solution

Answer

The truth value of \(\sim q \vee \sim p\) is true.

Step 1: Determine the Truth Value of p

To find out the truth value of statement p, calculate the left-hand side of the equation. Here, you add 4 + 6, which is 10. Since this matches with the right-hand side of the equation, statement p is therefore true.

Step 2: Determine the Truth Value of q

To find out the truth value of statement q, calculate the left-hand side of the equation. Here, you multiply 5 by 8, which is 40. Since this does not match the right-hand side of the equation, which is 80, statement q is false.

Step 3: Apply Logic Operations

First, apply the negation operator to both p and q. The negation of true is false, and the negation of false is true. So \(\sim p\) is false and \(\sim q\) is true. Then, apply the disjunction (or) operator to \(\sim p\) and \(\sim q\). This operator results in true if either of the operands (or both) is true. Since \(\sim q\) is true, \(\sim q \vee \sim p\) is also true.

Key Concepts

Logical Operations

Logical operations are fundamental tools in mathematics and computer science that are used to combine or modify statements, often referred to as propositions, to build compound statements and reveal logical relationships. In essence, logical operations follow specific rules, much like arithmetical operations, but instead of numerical values, they deal with truth values: true (T) and false (F).

The most commonly used logical operations include the logical AND (conjunction), logical OR (disjunction), and the logical NOT (negation). Understanding these operations allows us to process complex logical expressions and determine their overarching truth value based on their constituent parts. For example, in the given exercise, the negation and disjunction operations are applied to mathematical statements to deduce their combined truth value.

Negation Operator

The negation operator, often symbolized by a tilde (~) or the word 'not', plays a crucial role in logical expressions. This unary operator takes a single truth value as input and flips it; if the input is true (T), the output is false (F), and vice versa.

To apply this in mathematical statements, consider statement p, which is found to be true. Its negation \(\sim p\) is therefor false. Similarly, if we have a false statement, such as q in the exercise, the negation \(\sim q\) becomes true. Negation allows us to express contraries and contradictions, and it's crucial for constructing the logical opposite of a given proposition.

Disjunction Operator

The disjunction operator, represented by the symbol \(\vee\), corresponds to the logical 'or'. It's a binary operator that connects two statements and produces a true outcome if at least one of the two statements is true. The disjunction is inclusive, meaning that if both statements are true, the overall expression remains true.

In our exercise, we witness the disjunction at work when combining the negations of statements p and q. We find that \(\sim q \vee \sim p\) evaluates to true since at least one operand (in this case, \(\sim q\)) is true. The inclusive nature of disjunction is vital when considering possibilities and scenarios in various fields, from mathematics to decision-making.

Mathematical Statements

Mathematical statements are assertions that can definitively be said to be true or false. They are the building blocks of logical operations, as seen with statements p and q in the exercise. A statement such as \(4+6=10\) asserts a clear truth that can be proven by simple arithmetic. Similarly, \(5 \times 8=80\) asserts a claim that can be falsified through calculation.

Truth values for mathematical statements are the input for logical operations. Establishing whether these statements are true or false allows us to use logic operators effectively to arrive at new conclusions. It's the precision and clarity of mathematical statements that enable logical operations to be so powerful and widely applicable.