How can I prove it? (injection, bijection, surjection)

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SUMMARY

The discussion centers on proving that if the composition of two functions, g°f, is bijective, then function g must be surjective and function f must be injective. The user emphasizes the importance of understanding the definitions of bijective, surjective, and injective functions to construct a valid proof. The approach involves starting with the definition of a bijection and logically deducing the properties of g and f from this definition.

PREREQUISITES
  • Understanding of function composition, specifically g°f.
  • Knowledge of mathematical definitions: bijective, surjective, and injective functions.
  • Familiarity with basic proof techniques in mathematics.
  • Ability to visualize functions and their properties through diagrams.
NEXT STEPS
  • Study the definitions and properties of bijective, surjective, and injective functions in detail.
  • Learn how to construct mathematical proofs, focusing on function properties.
  • Explore examples of bijective functions and their implications in real-world applications.
  • Practice visualizing function compositions using diagrams to enhance understanding.
USEFUL FOR

Students studying mathematics, particularly those focusing on functions and proofs, as well as educators looking for clear explanations of function properties.

gotmejerry
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Homework Statement


How can I prove this?

If g°f is a bijective function, then g is surjective and f is injective.

Homework Equations





The Attempt at a Solution

 
Physics news on Phys.org
First what does g°f mean
what does bijective mean
what does surjective mean
what does injective mean
 
Start of with let g of f be a bijection, than state of the definition of a bijection. From there you can prove what must be true of g and f for g of f to meet the definition.
 
I can see why it is need to be true, when I draw it, unfotunately I cannot write down the solution in a mathematical way.
 
To start say
Let g°f be a bijective function.

then what can you say about g°f
 

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