Tautology and Contradiction |Fundamental Logic| S3 (2019)CSE & IT |Module 1-MAT203 | KTU BTech Part4

Introduction

This tutorial aims to explain the concepts of tautology and contradiction within the context of fundamental logic, as presented in the video from RVS Maths Academy. Understanding these concepts is crucial for students studying discrete mathematics, particularly in the fields of computer science and information technology.

Step 1: Understanding Tautology

Tautology refers to a statement that is always true, regardless of the truth values of its components.

• Definition: A tautological statement is one that yields a true result in every possible scenario.
• Example: The statement "It is raining or it is not raining" is a tautology because it covers all possibilities.
• Practical Tip: To identify a tautology, construct a truth table. If the final column (result) is always true, you have a tautology.

Creating a Truth Table for Tautology

1. List all variables involved in the statement.
2. Determine all possible combinations of truth values (True/False).
3. Evaluate the statement for each combination.
4. If the result is always true, the statement is a tautology.

Step 2: Understanding Contradiction

Contradiction refers to a statement that is always false, regardless of the truth values of its components.

• Definition: A contradictory statement yields a false result in every possible scenario.
• Example: The statement "It is raining and it is not raining" is a contradiction because it cannot be true at the same time.
• Common Pitfall: Misidentifying a complex statement as a contradiction when it may only be false under certain conditions.

Creating a Truth Table for Contradiction

1. List all variables involved in the statement.
2. Determine all possible combinations of truth values.
3. Evaluate the statement for each combination.
4. If the result is always false, the statement is a contradiction.

Step 3: Distinguishing Between Tautology, Contradiction, and Contingency

It is important to differentiate between tautologies, contradictions, and contingency (statements that can be either true or false).

• Contingency: A statement that can be true in some scenarios and false in others.
• Example: "It is raining" is a contingent statement; it could be true or false depending on the weather.

Tips for Differentiation

• Use truth tables to evaluate the truth values of statements.
• Remember that:
  • If the final truth table column is all true, it's a tautology.
  • If the final column is all false, it's a contradiction.
  • If the final column has both true and false values, it's a contingency.

Conclusion

In this tutorial, we explored the concepts of tautology and contradiction, key elements of fundamental logic in discrete mathematics. By utilizing truth tables, you can effectively identify these types of statements. As a next step, practice creating truth tables for various logical statements to reinforce your understanding. This foundational knowledge will aid you in more complex logical reasoning and applications in computer science and IT.