Math Problem Statement
Let f : A −→ B be a function. Show that it is not always the case that
f (f −1(Y )) = Y for all Y ⊆ B, by giving an explicit counterexample, but that the
equation is always true if f is surjective.
Solution
To show that does not always hold for all subsets , we can present a counterexample. Afterward, we'll demonstrate that this equation holds if is surjective.
Step 1: Counterexample
Let:
- Define a function by:
Here, is not surjective because there is an element that is not in the range of .
Now, take the subset .
-
Compute : Since and , neither of these values is . Therefore, no element in maps to , and thus:
-
Compute : Since , applying to yields:
-
Compare and : Here, .
Thus, we have a case where , which confirms that the equation does not always hold for arbitrary functions.
Step 2: Proof for Surjective Functions
Suppose is surjective, meaning that every element of has a preimage in . We want to show that for any subset , .
-
Show :
- For any , there exists some such that .
- Since , we have .
- Therefore, , which proves .
-
Show :
- Since is surjective, for each , there exists an such that .
- By the definition of , this is in .
- Thus, .
- This shows .
Since both inclusions hold, we conclude for all when is surjective.
Summary
- Counterexample: When is not surjective, we found .
- Proof for Surjectivity: If is surjective, for any subset .
Would you like further details or have any questions?
Related Questions
- How does the injectivity of affect the equation ?
- Can still hold if is injective but not surjective?
- What are the implications of for subsets when is injective?
- How would the equation change if were bijective?
- How can the preimage be used in probability theory or set theory?
Tip
When working with preimages and images in functions, carefully consider whether the function is surjective, as this often influences whether properties like hold.
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Math Problem Analysis
Mathematical Concepts
Set Theory
Functions
Surjectivity
Preimages and Images
Formulas
f(f^{-1}(Y))
Theorems
Properties of Surjective Functions
Suitable Grade Level
Undergraduate Mathematics