Problem 55 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

To accurately determine whether the statement is a tautology, a contradiction, or neither, the truth values of the final expression need to be examined after filling the truth table. The type of the statement will become apparent based on the results in the final column of the truth table.

Step 1: Setup the truth table

First, set up the truth table. It should contain a column for each variable (p, q, r), as well as columns for all the sub-expressions leading up the final full expression.

Step 2: Fill in the truth values

Start by filling in all the possible combinations of truth values for the variables p, q, and r. For each of these combinations, fill in the truth values for the sub expressions (\(q \rightarrow r\), \(r \longrightarrow \sim p\), and \(q \wedge p\)), using the definition of the logical connectors. Then fill in the truth values for the final full expression.

Step 3: Analyze the outcome

The final step is to analyze the outcome of the final full expression. If it contains only true values, the statement is a tautology. If it contains only false values, it is a self-contradiction. If it contains both true and false values, it is neither. Compare this outcome with the truth values of the sub expressions.

Key Concepts

Tautology

In logic, a tautology is a statement that is always true, regardless of the truth values of its individual parts. Imagine a statement as a machine with inputs and outputs, where the inputs are simple true or false conditions. No matter what combination of inputs you feed into this machine (true or false), if it always gives you a true outcome, then you have yourself a tautology. When analyzing complex logical statements, such as \( [(q \rightarrow r) \wedge(r \longrightarrow \sim p)] \leftrightarrow(q \wedge p) \), creating a truth table allows you to confirm whether the statement is indeed a tautology by breaking it down into simpler components and checking their outputs for every possible input combination.

If after filling the truth table, the column for the statement is all 'true', then it confirms that the statement is a tautology. This concept is foundational in logical analysis, as it helps to determine the strength and validity of arguments within mathematical reasoning, computer science, and philosophical debate.

Logical Connectors

Logical connectors, also known as logical operators, serve as the nuts and bolts of logical statements. They connect simple statements together to form more complex expressions. The most common logical connectors include AND (\(\wedge\)), OR (\(\vee\)), NOT (\(\sim\)), IF...THEN (\(\rightarrow\)) and IF AND ONLY IF (\(\leftrightarrow\)). Each connector has its own set of rules dictating how to combine truth values. For instance:

• AND (\(\wedge\)) yields true only if both connected statements are true.
• OR (\(\vee\)) yields true if at least one of the connected statements is true.
• NOT (\(\sim\)) yields the opposite truth value of the statement it precedes.
• IF...THEN (\(\rightarrow\)) yields false only if the first statement is true and the second is false.
• IF AND ONLY IF (\(\leftrightarrow\)) yields true if both statements are equally true or false.

By mastering these connectors, one can effectively construct and deconstruct logical statements, as well as interpret their meaning with precision.

Truth Values

The concept of truth values is central to the understanding of logic. Truth values are essentially the status of a statement being true or false. In logic, every statement can be assigned a truth value, and it's these values that logical connectors manipulate. When constructing a truth table, you list all possible combinations of truth values for the variables involved, and systematically apply the connectors to these values to get the truth value of more complex expressions.

For the logical expression given in the exercise, analyzing the truth values involves calculating the outcomes based on the connectors' rules. This step-by-step process reveals the logical behavior of the expression under different conditions, making truth tables an indispensable tool for anyone analyzing logical statements or designing digital logic circuits in computer engineering.