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January 5th, 2017, 10:23 AM   #1
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Find the values of a so that the operation is closed in [0;1]

I would appreciate anyone's contribution for the following exercises :

1)Find the values of a so that the operation

$\displaystyle x * y = x + y - xy - ax - ay +a $ is stable in the set [0;1]

//// First one was solved

2) For a=0, find a formula for $\displaystyle \overbrace{x*x*x*x* .....*x}^{n-times} $.

Last edited by MozartInACan; January 5th, 2017 at 10:46 AM.
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January 5th, 2017, 04:43 PM   #2
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Quote:
Originally Posted by MozartInACan View Post
I would appreciate anyone's contribution for the following exercises :

1)Find the values of a so that the operation

$\displaystyle x * y = x + y - xy - ax - ay +a $ is stable in the set [0;1]

//// First one was solved

2) For a=0, find a formula for $\displaystyle \overbrace{x*x*x*x* .....*x}^{n-times} $.
I assume that (2) is related to (1)

$x^n = (-1)^{n-1}x^n + \displaystyle{\sum_{k=1}^{n-1}}~(-1)^{k-1} \begin{pmatrix}n \\ k \end{pmatrix}x^k$

I leave it to you to prove this via induction.
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January 6th, 2017, 09:57 AM   #3
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When $a = 0$,
$\overbrace{x*x*x* \cdots *x}^{n\text{ terms}} = 1 - (1-x)^n$.

Last edited by quasi; January 6th, 2017 at 10:06 AM.
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