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(i) Consider a function of two numbers $F(a,b)$ that satisfies the condition $F(F(a,b),c)=F(a,b\cdot c)$.

For example $F$ could be one of the following:

$F(a,b) = a\cdot b$

$F(a,b) = a^b$

Question: are there any other functions that satisfy this equation (except from trivial ones such as $F=1$ etc.)? is it possible to find all of them ?

(ii) What are the functions that satisfy $F(F(a,b),c)=F(a,b^c)$ ?

Mohsen Shahriari
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user341440
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1 Answers1

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Solution of (ii): we have $F(F(a,b),1)=F(a,b)$ hence $F(x,1)=x$ (assuming that all $x$ belong to the image of $F$).

Therefore $F(F(a,1),c)=F(a,c)$. but also $F(F(a,1),c) = F(a,1^c)=F(a,1)=a$, so we find that

$F(a,b)=a$

is the only possible solution.

user341440
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