How are inverse images used to prove set inclusions?

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SUMMARY

The discussion focuses on proving set inclusions using inverse images in the context of functions. Specifically, it establishes that for a function f, the inclusion f-1(f(A)) ⊇ A holds true, and that f(f-1(B)) ⊆ B is also valid. The proof involves considering two cases for set A: when A is empty and when A is non-empty, employing the method of universal quantification to demonstrate the implications. The approach emphasizes the importance of assuming an arbitrary element x from set A to prove the necessary inclusions.

PREREQUISITES
  • Understanding of set theory, particularly the concepts of functions and inverse images.
  • Familiarity with universal quantification and logical implications.
  • Basic knowledge of proof techniques in mathematical analysis.
  • Experience with notation related to functions and sets, such as f(A) and f-1(B).
NEXT STEPS
  • Study the properties of inverse images in set theory.
  • Learn about the implications of functions in mathematical analysis.
  • Explore proof techniques involving universal quantifiers and logical reasoning.
  • Review examples of set inclusions and their proofs in the context of functions.
USEFUL FOR

Students of mathematical analysis, particularly those studying set theory and functions, as well as educators looking for clear examples of proof techniques involving inverse images.

drmarchjune
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I am studying the very first chapter of analysis, but can't quite get through this problem:

Prove f −1(f(A)) ⊇ A
Prove f(f −1(B)) ⊆ B
 
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How did you begin the problem?
For [itex]A \subseteq f^{-1}( f(A))[/itex], consider two cases.
case 1:[itex]A = \emptyset[/itex]
case 2: [itex]A \neq \emptyset[/itex]
 
in general , to prove that [tex]A\subseteq B[/tex] , we prove

[tex]\forall x[x\in A\Rightarrow x\in B][/tex]


and to prove this ,you let x be arbitrary. since we have an implication inside the square bracket, we assume antecedent , and set out to prove consequent . So assume
[tex]x\in A[/tex]
 

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