Elementary Symmetric Function/Examples/m = 1

Example of Elementary Symmetric Function: $m = 1$

Let $e_1 \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right)$ be an elementary symmetric function in $n$ variables of degree $1$.

Then:

$\ds e_1 \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right)$

$=$

$\ds x_1 + x_2 + \cdots + x_n$

By definition:

$\ds e_1 \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right)$

$=$

$\ds \sum_{1 \mathop \le j_1 \mathop \le n} \left({\prod_{i \mathop = 1}^1 x_{j_i} }\right)$

$\ds $

$=$

$\ds \sum_{1 \mathop \le j_1 \mathop \le n} x_{j_1}$

$\ds $

$=$

$\ds \sum_{j \mathop = 1}^n x_j$

$\ds $

$=$

$\ds x_1 + x_2 + \cdots + x_n$

$\blacksquare$