Composition of functions implies equality

In summary, we have three functions f:A->A, g:A->A, and h:A->A where f and g are bijective and h is bijective. We also know that f ° h = h ° g for every x in A. The question is whether f=g for every x in A. After attempting to solve it, a counterexample was found: A={1, 2, 3}, f={(1, 3), (2, 1), (3, 2)}, g={(1, 2), (2, 3), (3, 1)}, and h={(1, 3), (2, 2), (3, 1)}. This shows that f and
  • #1
netcaster
4
0
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 cannot find neither a counterexample nor a simple proof.
 
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  • #2
It took me just a few minutes to come up with a counter-example:

Let A= {1, 2, 3}, f= {(1,3), (2, 1), (3, 2)}, g= {(1, 2) , (2, 3), (3,1)}, h= {(1, 3), (2, 2), (3, 1)}.
 
  • #3
Many thanks! Your solution is simple and hence great!
 
  • #4
Your example is based on permutations..

What if h is only surjective rather than bijective?

The counterexample is not anymore valid...
 

FAQ: Composition of functions implies equality

What is the definition of "composition of functions"?

Composition of functions is the process of combining two or more functions to create a new function. The output of one function becomes the input of the next function, and this process continues until a final output is obtained.

How do you determine if two functions are equal using composition?

To determine if two functions are equal using composition, you need to plug the same value into both functions and see if they result in the same output. If the outputs are the same for all possible inputs, then the functions are equal.

Can functions of different types be composed?

Yes, functions of different types can be composed as long as the output of one function is in the domain of the next function. For example, you can compose a polynomial function with an exponential function.

What is the importance of understanding composition of functions?

Understanding composition of functions is important because it allows us to break down complex functions into simpler ones and solve problems more efficiently. It also helps us to understand the relationship between different functions and how they interact with each other.

Are there any restrictions when composing functions?

Yes, there are some restrictions when composing functions. The main restriction is that the output of one function must be in the domain of the next function. Additionally, the order in which the functions are composed can also affect the result, so it is important to pay attention to the order of operations.

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