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Suppose I have a set $\Omega = \{\omega_1,\omega_2,\omega_3,\ldots,\omega_k,\omega_n\}$ of $n$ elements. I ordered $\Omega$ with two different criteria by defining two new sets $A$ and $B$ of $k$ elements where $k$ is a fixed quantity $k<n$. Therefore $A$ and $B$ are two subsets. Suppose I want to define $C$ as a combination of the two subsets according a weight $\lambda \in [0,1]$, where if $\lambda=0$, $C=B$ and when $\lambda=1$, $C=A$. what is a proper definition of the set C?

Roughly explanation: What I want to do is the same reasoning behind a linear convex combination of two points $a$ and $b$ where $\lambda \in [0,1]$ , so $c = \lambda a + (1-\lambda)b$.

Can you help me? Thank you very much.

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Not every question involving sets has to do with set theory, and even less with the [set-theory] tag. – Asaf Karagila May 10 '12 at 21:24
up vote 0 down vote accepted

If $A=\{{\,a_1,a_2,\dots,a_k\,\}}$ and $B=\{{\,b_1,b_2,\dots,b_k\,\}}$ you could define $$C_{\lambda}=\{{\,\lambda a_1+(1-\lambda)b_1,\lambda a_2+(1-\lambda)b_2,\dots,\lambda a_k+(1-\lambda)b_k\,\}}$$ This assumes that when you wrote "I ordered $\Omega$ with two different criteria" you were using the word "ordered" the way I would use it, so that $A$ and $B$ are actually lists, rather than sets, and it makes sense to say which element of the one corresponds to any given element of the other.

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