Aljabar Abstrak - Pemetaan dan Jenis-jenisnya | Pemetaan Injektif, Surjektif, dan Bijektif

Introduction

This tutorial provides a comprehensive guide on abstract algebra, specifically focusing on mappings and their types, including injective, surjective, and bijective mappings. Understanding these concepts is essential for further studies in mathematics, especially in fields that deal with functions and relations.

Step 1: Understanding Mappings

• Definition: A mapping (or function) relates elements from one set to another set.
• Notation: Typically represented as f: A → B, where A is the domain and B is the codomain.
• Key Point: Each element in set A is associated with exactly one element in set B.

Practical Tips

• Visualize mappings with arrows connecting elements from the domain to the codomain to understand their relationships better.

Step 2: Exploring Types of Mappings

Mappings can be categorized into three types: injective, surjective, and bijective.

Step 2.1: Injective Mappings

• Definition: An injective mapping (or one-to-one mapping) occurs when different elements in the domain map to different elements in the codomain.
• Condition: For any x1, x2 in A, if f(x1) = f(x2), then x1 must equal x2.

Example

• Consider f: {1, 2, 3} → {A, B, C} where f(1) = A, f(2) = B, f(3) = C. This is injective because no two elements in the domain map to the same element in the codomain.

Common Pitfall

• Confusing injective with surjective; remember that injective focuses on distinct outputs for distinct inputs.

Step 2.2: Surjective Mappings

• Definition: A surjective mapping (or onto mapping) occurs when every element in the codomain has at least one element from the domain mapping to it.
• Condition: For every b in B, there exists at least one a in A such that f(a) = b.

Example

• Consider f: {1, 2, 3} → {A, B} where f(1) = A, f(2) = A, f(3) = B. This is surjective because both A and B in the codomain have corresponding elements in the domain.

Practical Tips

• Use Venn diagrams to visualize surjective mappings, showing how the entire codomain is covered.

Step 2.3: Bijective Mappings

• Definition: A bijective mapping is both injective and surjective, meaning each element in the domain maps to a unique element in the codomain and covers the entire codomain.

Example

• Consider f: {1, 2, 3} → {A, B, C} where f(1) = A, f(2) = B, f(3) = C. This is bijective because it is both one-to-one and onto.

Real-World Applications

• Bijective mappings are crucial in cryptography and coding theory, where a one-to-one correspondence is needed for secure communication.

Conclusion

In this tutorial, we covered the essential concepts of mappings in abstract algebra, focusing on injective, surjective, and bijective types. Understanding these mappings is fundamental for advanced mathematical studies and their applications in various fields.

Next Steps

• Practice identifying and constructing examples of each type of mapping.
• Explore more complex functions and their properties in algebra.