Elementary Symmetric Function/Examples/m = 0

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.

$\blacksquare$