Math Problem Statement
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 and , and let (identity function on ).
- Show that is one-to-one (injective) and 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 and , such that their composition is the identity function on , denoted by . This means for all , .
- To show:
- is injective (one-to-one).
- is surjective (onto).
Hints for Solution:
-
Proving that is injective:
- If for , apply to both sides. You will use the fact that to demonstrate that , which implies that is injective.
-
Proving that is surjective:
- Since , for every , there is a corresponding , such that . This suggests that maps some element of to every element of , helping establish that 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:
- What is the difference between injective and surjective functions?
- How can you prove that a function is injective?
- What is the significance of the identity function in function composition?
- Can a function be bijective if it is not injective or surjective?
- 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)
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