Problem 1 - 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. \(\sim q\)

Step-by-Step Solution

Answer

The truth value of the statement \(\sim q\), is true.

Step 1: Identify the statements

This exercise involves two statements known as (\(p\) and \(q\)). Identify the statement associated with \(q\), which is, '5 x 8 = 80'.

Step 2: Evaluate the truth value of \(q\)

Evaluate whether the mathematical equation in \(q\) is true or false. Checking the equivalency, \(5 \times 8 = 40\), not 80. So, statement \(q\) is false.

Step 3: Determine the truth value for \(\sim q\)

The symbol \(\sim\) denotes logical negation. In other words, if something is true, the negation is false, and if something is false, the negation is true. As \(q\) is false, then, the negation of \(q\), \(\sim q\), must be true.

Key Concepts

Truth Values

In the world of logic, statements are usually characterized as either true or false. This binary distinction is what we refer to as "truth values." In mathematics, resolving the truth value of a statement involves checking specific propositions to see if they hold true in reality. Let's break it down further:

• True: If a statement matches a proven or established fact, its truth value is "true." For example, the statement "4 + 6 = 10" is true because mathematically, it sums up correctly.


• False: Conversely, when a statement does not align with a factual situation, its truth value is "false." A statement like "5 × 8 = 80" does not match the actual result of the multiplication, which would be 40, making it false.

Understanding truth values is crucial, especially when dealing with more complex logical operations or statements, as it is the foundation on which logical reasoning is built.

Logical Negation

Logical negation is a powerful operation in propositional logic. It deals with reversing the truth value of a proposition. Basically, it is the "not" operator in logical terms. The symbol for logical negation is "\(\sim\)," and it is applied to propositions to revert their truth values.

• If a statement, say, "p," is true, then applying a logical negation to it (\(\sim p\)) would make it false.


• If a statement is false, then its negation would be true.

This operation is crucial for forming new propositions from existing ones and is fundamental in developing logical conditions. For example, if our statement \("q"\) is false, the logical negation \(\sim q\) would be true. Logical negation helps in the creation of more complex logical expressions and the understanding of logical frameworks.

Propositional Logic

Propositional logic is a branch of logic that studies propositions and their components. It is concerned with statements that can be assigned a truth value, either true or false. The essence of propositional logic lies in how these basic and compound statements interact through logical connectives.

• Statements: These are simple assertions that can hold a truth value. Each proposition is a statement like "\(p\)" or "\(q\)."


• Connectives: Logical connectives such as "and," "or," "not," "if...then..." help us build complex statements by combining simple propositions.


• Applications: It's used in computer science, mathematics, and philosophical logic to develop arguments, design algorithms, and reason about conditions.

With propositional logic, reasoning becomes systematic, allowing us to deduce new truths from a set of known truths. Understanding how propositions link together can aid in tackling complex problems by breaking them down into simpler, more manageable parts.