Problem 48 - Thinking Mathematically

Question

Use a truth table to determine whether each statement is a tautology, a self- contradiction, or neither. \((p \rightarrow q) \leftrightarrow(p \vee \sim q)\)

Step-by-Step Solution

Answer

By following the steps and constructing the truth table, it can be observed that the statement \((p \rightarrow q) \leftrightarrow(p \vee \sim q)\) is neither a tautology nor a self-contradiction. It results in a mixture of 'true' and 'false' values.

Step 1: Define the structure of the truth table

Given the statement \((p \rightarrow q) \leftrightarrow(p \vee \sim q)\), start by creating a truth table with columns for \(p\), \(q\), \(\sim q\), \(p \rightarrow q\), \(p \vee \sim q\), and \((p \rightarrow q) \leftrightarrow(p \vee \sim q)\).

Step 2: Populate truth values

Populate the truth table with all possible combinations of true and false for \(p\) and \(q\), that is, both can be either true (T) or false (F). This results in 4 scenarios to consider.

Step 3: Compute values for each part of the statement

Next, compute the values for \(\sim q\), \(p \rightarrow q\) and \(p \vee \sim q\) based on the values of \(p\) and \(q\). For \(\sim q\) replace 'true' with 'false' and vice versa. For \(p \rightarrow q\), the result is 'false' only when \(p\) is true and \(q\) is false, otherwise it is 'true'. For \(p \vee \sim q\), the result is 'true' if either \(p\) is 'true' or \(\sim q\) is 'true', otherwise it is 'false'.

Step 4: Evaluate the final result

Evaluate \((p \rightarrow q) \leftrightarrow(p \vee \sim q)\) based on the results from Step 3. This result is 'true' if both \(p \rightarrow q\) and \(p \vee \sim q\) have the same truth values, otherwise it is 'false'.

Step 5: Conclude

Examine the results in the last column. If all 'true', then the statement is a tautology. If all 'false', then it is a self-contradiction. If it is a mix of 'true' and 'false', then it is neither.

Key Concepts

Tautology in Mathematics

A tautology in the realm of mathematics, especially when referring to logic, is a statement that is always true, regardless of the truth values of its constituent variables. In simpler terms, no matter what the scenario, a tautological statement cannot be anything but true.

For instance, the statement 'It is raining or it is not raining' is a tautology because, whether it rains or not, the statement holds true. In the context of truth tables, a tautology is identified when the final column—after evaluating all possibilities—consists of nothing but 'true' entries. As tautologies are fundamental to logical proofs, their identification is crucial in validating arguments and propositions in various fields of logic and computer science.

Logical Equivalence

Logical equivalence is a significant concept in mathematics which refers to two statements that have the same truth values in all possible scenarios. When you create a truth table, the columns for two logically equivalent statements will be identical. This is like saying two different roads always lead to the same destination.

In practice, logical equivalence allows mathematicians and logicians to simplify complex expressions and prove that different statements effectively say the same thing. For example, the expressions \((p \rightarrow q)\) and \((\sim p \vee q)\) are logically equivalent because if you replace 'p' and 'q' with any true or false values, the results will align perfectly in a truth table. If you can swap one statement for another without altering the overall meaning or outcome, those statements are logically equivalent. This is crucial for simplifying logical expressions and solving logical dilemmas.

Conditional Statements

In the realm of logic, a conditional statement typically follows the 'if-then' format, formally written as \((p \rightarrow q)\). It denotes that if the first part, \((p)\), known as the 'hypothesis', is true, then the second part, \((q)\), known as the 'conclusion', must also be true.

However, if the 'hypothesis' is false, the 'conclusion' can be either true or false, and the whole conditional statement would still be considered true. This can be counterintuitive at first glance but is essential in constructing definitions, theorems, and logical proofs. Understanding truth tables helps students to comprehend how the truth value of the whole statement depends on the interaction between its parts.