Problem 21 - Discrete Mathematics with Applications

Question

Let \(t\) be a tautology and \(p\) an arbitrary proposition. Give the truth value of each proposition. $$\sim t \wedge p$$

Step-by-Step Solution

Answer

The compound proposition \(\sim t \wedge p\) is always false (F) regardless of the truth value of the arbitrary proposition \(p\).

Step 1: Understanding tautology and arbitrary proposition

First, we need to understand the difference between a tautology and an arbitrary proposition. A tautology is a statement that is always true, no matter the truth values of its components. In this case, we are given the symbol \(t\) to represent a tautology, which means that the truth value of \(t\) is always true (T). An arbitrary proposition is a statement that can be either true or false, depending on the context or information provided. In this exercise, we have \(p\) as an arbitrary proposition, which means its truth value can be either true (T) or false (F).

Step 2: Apply Negation operation on tautology

The given compound proposition has the negation operation \((\sim)\) applied to the tautology \(t\). We already know that the truth value of a tautology is true, so when we apply the negation operation, it simply negates the truth value of \(t\). So, \(\sim t\) would have a truth value of false (F).

Step 3: Apply Conjunction operation on negated tautology and arbitrary proposition

The final operation in the compound proposition is the conjunction operation \((\wedge)\), which combines the negated tautology \(\sim t\) and the arbitrary proposition \(p\). The conjunction operation has the following truth table: \(p \ \ \ \ \ q \ \ \ \ \ p \wedge q\) \\ T \ \ \ \ \ T \ \ \ \ \ T \\ T \ \ \ \ \ F \ \ \ \ \ F \\ F \ \ \ \ \ T \ \ \ \ \ F \\ F \ \ \ \ \ F \ \ \ \ \ F \\ In our case, since \(\sim t\) is always false, we have: \(\sim t \ \ \ \ \ p \ \ \ \ \ \sim t \wedge p\) \\ F \ \ \ \ \ \ \ \ \ T \ \ \ \ \ \ \ \ F \\ F \ \ \ \ \ \ \ \ \ F \ \ \ \ \ \ \ \ F \\

Step 4: Final Result

From the above evaluation, we can see that the compound proposition \(\sim t \wedge p\) is always false (F) regardless of the truth value of the arbitrary proposition \(p\).

Key Concepts

Propositional Logic

In any logical system, including mathematics and computer science, propositional logic serves as the foundation. It's a branch of logic dealing with propositions, which are statements that can be either true or false. Unlike predicates, propositional logic doesn't concern itself with the internal structure of statements. Instead, it focuses on the truth values assigned to them. Propositions can be simple, like "It is raining," or complex, involving several smaller propositions combined using logical connectives.

• Atomic: A basic statement with no logical connectives, e.g., "The sky is blue."
• Compound: A statement formed by combining multiple atomic propositions using logical connectives, such as "and," "or," and "not."

Propositional logic comes in handy when evaluating logical arguments and constructing truth tables. It helps to assess the overall truth values of compound propositions by breaking them down into atomic components.

Negation

Negation is a fundamental logical operation within propositional logic. It involves flipping the truth value of a given proposition. If a statement is true, its negation is false, and vice versa. The negation of a proposition \(p\) is often represented as \(\sim p\) or sometimes using the symbol "¬".
Understanding negation is critical when working with more complex logical formulas. For instance:

• If \(p\) is "Today is sunny," the negation \(\sim p\) would be "Today is not sunny."
• In truth tables, negation will invert the row's truth value. If \(p\) is true (T), \(\sim p\) becomes false (F), and if \(p\) is false (F), \(\sim p\) turns true (T).

Negation is often used in conjunction with other logical operations to form more complex expressions, as seen in this exercise with its application on the tautology \(t\).

Conjunction

Conjunction is another key operation in propositional logic. It is represented by the symbol \(\wedge\), meaning "and." When two propositions are combined using \(\wedge\), the resulting compound proposition is true only if both individual propositions are true. Otherwise, the compound proposition is false.
Here's an example:

• If \(p\) is "It is raining" and \(q\) is "I have an umbrella," then the conjunction \(p \wedge q\) means "It is raining and I have an umbrella."
• In a truth table, only the combination \(T \wedge T\) results in true (T); all other combinations (\(F \wedge T\), \(T \wedge F\), and \(F \wedge F\)) yield false (F).

Understanding conjunction helps determine the outcome of compound propositions, such as \(\sim t \wedge p\) in the provided exercise, where one component is always false.

Truth Tables

Truth tables are a powerful tool in propositional logic used to represent the truth values of propositions and their combinations. They list all possible scenarios for the propositions and show how the truth value of a compound proposition is determined by its components.
Using truth tables, we can systematically explore every possible truth value for atomic propositions and their logical combinations.

• They include columns for each proposition and each operation involved in the formula.
• For the expression \(\sim t \wedge p\), the table lists the possible values of \(p\) and shows \(\sim t\) as always false (F) due to the negation of the tautology.

Truth tables help visualize the logical structure and evaluate expressions calmly and precisely. They're beneficial in determining the outcomes of logical operations, as seen by concluding that the compound proposition \(\sim t \wedge p\) is always false in this exercise.