Composition Of Functions Implies Equality

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SUMMARY

The discussion centers on the equality of two bijective functions, f and g, under the condition that f ° h = h ° g for all x in the set A. The participants explore whether this implies that f equals g for every x in A. A specific example is provided with the set A = {1, 2, 3}, where f, g, and h are defined as bijective mappings. The consensus leans towards the conclusion that f must equal g, although no definitive proof or counterexample has been established.

PREREQUISITES
  • Understanding of bijective functions
  • Knowledge of function composition
  • Familiarity with set theory
  • Basic principles of mathematical proof techniques
NEXT STEPS
  • Study the properties of bijective functions in detail
  • Learn about function composition and its implications
  • Explore mathematical proof techniques, particularly in set theory
  • Investigate counterexamples in function equality scenarios
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone interested in the properties of functions and their compositions.

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We have three functions: f:A->A, g:A->A and h:A->A
with both f and g bijective and h bijective.

We know that f ° h = h ° g for every x in A.

Is it true that f=g for every x in A?

I have tried to solve it and I am pretty sure it is true but I can find neither a counterexample nor a simple proof.
 
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Take A= {1, 2, 3}, f= {(1,3), (2,1), (3,2)}, g= {(1,2), (2,3), (3,1)}, h= {(1,3), (2,3), (3, 1)}.
 

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