Proving the graph of a composite function

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SUMMARY

The discussion focuses on proving that the set Γ, defined as θ(Γf × C ∩ A × Γg), represents the graph of the composite function g◦f. The participants emphasize the need to demonstrate that the first projection of Γ is a bijection. The solution involves understanding the projections and intersections of the graphs of the functions f and g, specifically Γf and Γg, within the context of set theory and function composition.

PREREQUISITES
  • Understanding of set theory, particularly Cartesian products and intersections.
  • Knowledge of function composition and the concept of graphs of functions.
  • Familiarity with projections in the context of multi-dimensional sets.
  • Basic skills in mathematical proofs and bijections.
NEXT STEPS
  • Study the properties of Cartesian products in set theory.
  • Learn about function composition in detail, focusing on bijective functions.
  • Explore the concept of projections in higher-dimensional spaces.
  • Review examples of proving properties of graphs of functions in mathematics.
USEFUL FOR

Mathematics students, educators, and anyone interested in advanced topics in set theory and function analysis.

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Homework Statement



Let f : A → B, g : B → C be functions where A,B,C are sets. ConsiderΓf ⊂A×B,the graph of f,Γg ⊂B×C,the graph of g. Now consider the sets Γ f ×C ⊂ A×B×C and A×Γg ⊂A×B×C. LetΓ=θ(Γf ×C∩A×Γg)⊂A×C where θ : A×B×C → A×C is the projection defined as θ((a,b,c))=(a,c). Show that Γ is the graph of g◦f.

Homework Equations





The Attempt at a Solution



I think that I should find a way to show that the first projection of gamma is a bijection. But i don't know how I can do that.
 
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please help I've been trying for so long
 
Then show us what you have been doing all that time!
 

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