Can an Injection Prove Equality in the Intersection of Subsets in a Function?

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SUMMARY

The discussion centers on proving that for an injective function f: A → B, the equality f(A1 ∩ A2) = f(A1) ∩ f(A2) holds true. Participants confirm that while the inclusion f(A1 ∩ A2) ⊆ f(A1) ∩ f(A2) is established, the challenge lies in demonstrating the reverse inclusion. The key to resolving this is to show that if y is in f(A1) ∩ f(A2), then y must also be in f(A1 ∩ A2), leveraging the properties of injective functions.

PREREQUISITES
  • Understanding of set theory, specifically intersections and subsets.
  • Knowledge of functions, particularly the definition and properties of injective functions.
  • Familiarity with the concept of proving set equality through subset relationships.
  • Basic mathematical notation and logic used in proofs.
NEXT STEPS
  • Study the properties of injective functions in detail to understand their implications on set mappings.
  • Learn about proving set equality through subset inclusion, focusing on both directions of the proof.
  • Explore examples of functions and their images to solidify understanding of intersections in set theory.
  • Investigate related concepts such as surjective and bijective functions for a broader perspective on function properties.
USEFUL FOR

Mathematicians, computer scientists, and students studying discrete mathematics or set theory, particularly those interested in function properties and proofs.

ragnes
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f: A-->B is a function. A,B are sets.
Let A1, A2 be contained in/equal to A.

f(A1 intersection A2) is contained in OR equal to f(A1) intersection with f(A2). Show that the equality holds if f is an injection.

I know how to prove that it is contained, but not the equal/injection part. Help please!
 
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ragnes said:
f: A-->B is a function. A,B are sets.
Let A1, A2 be contained in/equal to A.

f(A1 intersection A2) is contained in OR equal to f(A1) intersection with f(A2). Show that the equality holds if f is an injection.

I know how to prove that it is contained, but not the equal/injection part. Help please!

Okay, so far you have proved that...
f(A_{1}\cap A_{2}) \subseteq f(A_{1})\cap f( A_{2})

When you want to prove equality of sets ( say A and B) you have to show A\subset B and B \subset A.

Where in the process of showing equality are you stuck ?
 

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