Problem 48 - Discrete Mathematics with Applications

Question

Let \(t\) be a tautology and \(p\) an arbitrary proposition. Find the truth value of each. $$(p \vee t) \rightarrow t$$

Step-by-Step Solution

Answer

The truth value of the compound proposition \((p \vee t) \rightarrow t\) is always true, regardless of the truth value of the arbitrary proposition \(p\).

Step 1: Analyze given propositions

We are given a tautology \(t\) and an arbitrary proposition \(p\). A tautology is a proposition that is always true, regardless of the truth values of its constituent propositions. In this case, the truth value of \(t\) is true. An arbitrary proposition \(p\) can be either true or false; we will consider both possibilities.

Step 2: Create a truth table

To find the truth value of the compound proposition \((p \vee t) \rightarrow t\), we will create a truth table considering all possible truth values for \(p\). Since \(t\) is always true, there are only two possibilities for the truth value of \(p\) - true and false. | \(p\) | \(t\) | \(p \vee t\) | \((p \vee t) \rightarrow t\) | |-----|-----|------|-------| | T | T | T | T | | F | T | T | T |

Step 3: Analyze the truth table

Looking at the truth table, we can see that for all possible truth values of \(p\), the compound proposition \((p \vee t) \rightarrow t\) has a truth value of true. So, the truth value of the given compound proposition \((p \vee t) \rightarrow t\) is true, regardless of the truth value of the arbitrary proposition \(p\).

Key Concepts

Truth Table

Truth tables are a fundamental tool in propositional logic, making abstract concepts easier to understand through visual representation. A truth table lists all possible truth values of given propositions, along with the results of those values for a particular logical expression. For example, when evaluating

• \((p \vee t) \rightarrow t\), where \(t\) is always true, the truth table helps us visualize how the compound proposition behaves.
• To create a truth table for our expression, list all possible truth values of \(p\). Since \(t\) is a tautology, it's always true. Hence, there are just two possibilities: \(p\) being true (T) or false (F).

Observe how each truth value combination affects the compound proposition. You'll find that regardless of whether \(p\) is true or false, the result of the expression remains true. This visualization confirms the logical deduction that the compound proposition is indeed a tautology itself.

Compound Proposition

A compound proposition is formed by combining two or more simpler propositions using logical connectors such as "and" (\(\land\)), "or" (\(\lor\)), and "implies" (\(\rightarrow\)). These connectors help determine the overall truth value of the compound statement based on its components. In the example compound proposition

• \((p \vee t)\) and \((p \vee t) \rightarrow t\), the connector \(\lor\) is used in the first part, representing an 'or' operation, which results in true if at least one of \(p\) or \(t\) is true.
• Meanwhile, the implication operator \(\rightarrow\) correlates the results, implying that if the left side is true, the right side must also be true for the whole expression to be true.

This logical combination is what makes compound propositions interesting and challenging, as they encapsulate more complex truth scenarios.

Propositional Logic

Propositional logic forms the backbone of logical reasoning in mathematics and computer science. It's a branch of logic that deals with propositions, which are statements that can either be true or false. Propositions are combined to form compound propositions using logical operations. The basic logical operations include

• "and" (\(\land\)), "or" (\(\lor\)), and "not" (\(eg\)),
• "implies" (\(\rightarrow\)), which evaluates to false only when a true proposition leads to a false one.

Understanding and employing these basic operations allows us to deduce, infer, and make logical arguments. For instance, in the expression \((p \vee t) \rightarrow t\), propositional logic dictates that because \(t\) is always true, the expression is unaffected by \(p\)'s variability, hence always true. Thus, propositions and their combinations, governed by propositional logic, are key to assessing complex logical statements.