Logika Matematika, Kalimat Majemuk Tautologi, Kontradiksi dan Kontingensi

Introduction

This tutorial explores the concepts of tautologies, contradictions, and contingencies in mathematical logic. Understanding these terms is essential for evaluating logical statements and their truth values. In this guide, we will provide definitions, examples, and a step-by-step method to analyze logical statements using truth tables.

Step 1: Understand Key Concepts

Before diving into examples, familiarize yourself with the definitions of tautology, contradiction, and contingency.

• Tautology: A compound statement that is always true, regardless of the truth values of its components.
• Contradiction: A compound statement that is always false for any truth values of its components.
• Contingency: A compound statement that can be true or false, depending on the truth values of its components.

Step 2: Create Truth Tables

To analyze logical statements, create truth tables. A truth table lists all possible truth values of the components and the resulting value of the compound statement.

How to Create a Truth Table

1. Identify the components of the statement (e.g., p, q, r).
2. List all possible combinations of truth values (True or False) for these components.
3. Calculate the truth value of the compound statement for each combination.

Example Truth Table

For the statement p → (p ∨ q):

| p | q | p ∨ q | p → (p ∨ q) | |-------|-------|-------|-------------| | True | True | True | True | | True | False | True | True | | False | True | True | True | | False | False | False | True |

Step 3: Analyze Specific Statements

Now, let’s evaluate specific statements to classify them as tautologies, contradictions, or contingencies. Use the truth table method for each of the following statements:

1. p → (p ∨ q)
2. (p ∧ q) ∧ (p → ~q)
3. (p ∨ q) → r
4. ((p ∨ q) ∧ ~p) → q
5. (p ∧ q) → ~p
6. (p → q) ∧ p
7. ~(p → q) ∧ ~p
8. ((p → q) ∧ p) → q
9. (p ∧ q) → (q ∨ r)
10. p ∧ (~p ∧ q)

Example Analysis

Let’s analyze the first statement p → (p ∨ q).

• From the truth table, you can see that the result is always True. Hence, it is classified as a tautology.

Continue the same process for the remaining statements.

Conclusion

In this tutorial, we covered the fundamental concepts of tautologies, contradictions, and contingencies, and provided a systematic approach to analyze logical statements using truth tables. By understanding these concepts, you can better evaluate logical expressions in mathematics and computer science.

Next Steps

• Practice creating truth tables for different logical statements.
• Explore more complex logical expressions and their evaluations.
• Consider studying related topics such as logical equivalence and quantifiers for a deeper understanding of mathematical logic.