Скачать или смотреть Lec 30: Mastering Surjective (Onto) Functions: Definitions, Infinite Sets, and Solved Examples

Following our discussion on One-to-One functions, today’s session explores Surjective functions, commonly known as Onto functions. We break down the formal definition, visualize mappings, and learn the rigorous algebraic method required to prove surjectivity on infinite sets like ℝ and ℤ.

📖 What You Will Learn:
Core Definition: A function f: X → Y is surjective if every element in the codomain (Y) is mapped to at least once by an element in the domain (X).

The Codomain Rule: In an onto function, every element of the co-domain is the image of some elements of its domain.

The Proof Strategy: A step-by-step breakdown of the formal proof method:

Suppose y (or m) is an arbitrary element of the codomain.

Show that there exists an element x (or n) in the domain such that f(x) = y.

Solve for the domain variable in terms of the codomain variable and check if it belongs to the defined domain set.

💡 Solved Examples Included:
Proving a Function is Onto: A detailed look at f(x) = 4x - 1 where the domain and codomain are the set of Real Numbers (ℝ).

Identifying Non-Surjective Functions: Analyzing h(n) = 4n - 1 over the set of Integers (ℤ) to understand why it fails to be onto when the result isn't an integer (n ∉ ℤ).

🔢 Key Symbols Used (Unicode):
∀ — For all
∃ — There exists
∈ — Element of
∉ — Not an element of
ℝ — Real Numbers Set
ℤ — Integers Set
→ — Maps to