How Can Injectivity Prove f^(-1)(f(A)) = A?

  • Context: Undergrad 
  • Thread starter Thread starter m_kosak
  • Start date Start date
  • Tags Tags
    Function Injective
Click For Summary
SUMMARY

The discussion centers on proving the statement f^(-1)(f(A)) = A under the condition that the function f: X → Y is injective. The key argument is based on demonstrating two inclusions, starting with f^(-1)(f(A)) ⊆ A. The injectivity of f ensures that if f(a) = f(b), then a must equal b, which is crucial for establishing the required inclusion. The proof requires showing that any element x in f^(-1)(f(A)) must also belong to the subset A.

PREREQUISITES
  • Understanding of injective functions in mathematics
  • Familiarity with set theory and subset notation
  • Knowledge of function notation and inverse functions
  • Basic proof techniques, particularly proof by inclusion
NEXT STEPS
  • Study the properties of injective functions in detail
  • Learn about set operations and their implications in proofs
  • Explore examples of proving function properties using inclusions
  • Review the concept of inverse functions and their applications
USEFUL FOR

Mathematics students, educators, and anyone interested in understanding function properties and proofs related to injectivity and set theory.

m_kosak
Messages
1
Reaction score
0
who can help me?
ı want to prove this
If f : X → Y is injective and A is a subset of X, then f −1(f(A)) = A.
but how can I do this :(
 
Physics news on Phys.org
So being injective means that whenever f(a) = f(b) in Y, then a = b in X.

The usual proof for such statements is to show two inclusions. Let's start with [itex]f^{-1}(f(A)) \subseteq A[/itex].
Let x be an element of the set on the left hand side. So x is an element in X, for which [itex]x \in f^{-1}(f(A))[/itex]. Can you show that x should in fact be an element of A?
 

Similar threads

Undergrad On injectivity of two-sided Laplace transform
  • · Replies 5 ·
Replies
5
Views
3K
MHB Why is f being an injection equivalent to this ?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Proof involving functional graphs and the injective property
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Cardinality of decreasing functions from N to N
  • · Replies 19 ·
Replies
19
Views
4K
MHB Prove Injectivity & Surjectivity of Composite Application f
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Injective immersion and embedding
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Negation of statement involving cardinalities
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad Is √9x a Bijection from N to R?
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad How do E[X] and E[|X|] relate?
  • · Replies 6 ·
Replies
6
Views
2K
Graduate Show a functions inverse is injective iff f is surjective
  • · Replies 10 ·
Replies
10
Views
4K