Elementary Symmetric Function/Examples/m = 0
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Example of Elementary Symmetric Function: $m = 0$
Let $\map {e_0} {\set {x_1, x_2, \ldots, x_n} }$ be an elementary symmetric function in $n$ variables of degree $0$.
Then:
- $\map {e_0} {\set {x_1, x_2, \ldots, x_n} } = 1$
Proof
By definition:
\(\ds \map {e_0} {\set {x_1, x_2, \ldots, x_n} }\) | \(=\) | \(\ds \sum_{1 \mathop \le n} \paren {\prod_{i \mathop = 1}^0 x_{j_i} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{1 \mathop \le n} \paren 1\) | Definition of Vacuous Product | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
The validity of the material on this page is questionable. In particular: $\sum_{1 \mathop \le n} 1 = \paren 1$? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Questionable}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Whether the summation $\ds \sum_{1 \mathop \le n}$ makes sense, as such, is a moot point.
$\blacksquare$