Composition Of Functions Implies Equality

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The discussion revolves around the relationship between three bijective functions f, g, and h defined on the same set A. It is established that if f ° h = h ° g for every x in A, the question arises whether this implies f = g for all x in A. A user presents a specific example with A = {1, 2, 3} and defines f, g, and h, seeking to prove or disprove the equality of f and g. Despite attempts to find a counterexample or a straightforward proof, the conclusion remains uncertain. The thread highlights the complexity of function composition and the implications of bijectivity in this context.
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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)}.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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