Math Problem Statement

Exercise 3.3.12. Suppose f: X → Y and g: Y → Z, and let g ∘ f = Id_X (identity function on X). Show that f is injective (one-to-one) and g is surjective (onto).

Solution

The image contains a portion of a mathematical exercise from a textbook. Based on the visible text, it appears to be related to the properties of functions, particularly bijections, and function compositions.

The text reads:

  • Exercise 3.3.12. Suppose f:XYf: X \to Y and g:YZg: Y \to Z, and let gf=IdXg \circ f = Id_X (identity function on XX).
    • Show that ff is one-to-one (injective) and gg is onto (surjective).

This is a typical problem involving function composition and properties like injectivity (one-to-one) and surjectivity (onto). Let's break down the task of proving the statements.

Problem Breakdown:

You are given two functions f:XYf: X \to Y and g:YZg: Y \to Z, such that their composition gfg \circ f is the identity function on XX, denoted by IdXId_X. This means for all xXx \in X, g(f(x))=xg(f(x)) = x.

  • To show:
    1. ff is injective (one-to-one).
    2. gg is surjective (onto).

Hints for Solution:

  1. Proving that ff is injective:

    • If f(x1)=f(x2)f(x_1) = f(x_2) for x1,x2Xx_1, x_2 \in X, apply gg to both sides. You will use the fact that gf=IdXg \circ f = Id_X to demonstrate that x1=x2x_1 = x_2, which implies that ff is injective.
  2. Proving that gg is surjective:

    • Since gf=IdXg \circ f = Id_X, for every xXx \in X, there is a corresponding f(x)Yf(x) \in Y, such that g(f(x))=xg(f(x)) = x. This suggests that gg maps some element of YY to every element of XX, helping establish that gg is surjective.

Let me know if you'd like a detailed step-by-step solution, or if you have any specific questions!

5 Related Questions:

  1. What is the difference between injective and surjective functions?
  2. How can you prove that a function is injective?
  3. What is the significance of the identity function in function composition?
  4. Can a function be bijective if it is not injective or surjective?
  5. How do function compositions affect the properties of injectivity and surjectivity?

Tip: When working with function compositions, always keep in mind how the properties of the individual functions affect the overall composition.

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Math Problem Analysis

Mathematical Concepts

Functions
Injectivity
Surjectivity
Composition of Functions

Formulas

g ∘ f = Id_X
Injective: f(x₁) = f(x₂) → x₁ = x₂
Surjective: ∀ z ∈ Z, ∃ y ∈ Y such that g(y) = z

Theorems

Identity function property
Injectivity criterion
Surjectivity criterion

Suitable Grade Level

Undergraduate (Mathematics)