Elementary Symmetric Function/Examples/m = 1

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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 $1$.


\(\ds e_1 \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right)\) \(=\) \(\ds x_1 + x_2 + \cdots + x_n\)


By definition:

\(\ds e_1 \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right)\) \(=\) \(\ds \sum_{1 \mathop \le j_1 \mathop \le n} \left({\prod_{i \mathop = 1}^1 x_{j_i} }\right)\)
\(\ds \) \(=\) \(\ds \sum_{1 \mathop \le j_1 \mathop \le n} x_{j_1}\)
\(\ds \) \(=\) \(\ds \sum_{j \mathop = 1}^n x_j\)
\(\ds \) \(=\) \(\ds x_1 + x_2 + \cdots + x_n\)