Elementary Symmetric Function/Examples/m = 2
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Example of Elementary Symmetric Function: $m = 2$
Let $e_2 \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right)$ be an elementary symmetric function in $n$ variables of degree $2$.
Then:
| \(\ds e_2 \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right)\) | \(=\) | \(\ds x_1 x_2 + x_1 x_3 + \cdots + x_1 x_n\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds x_2 x_3 + \cdots + x_2 x_n\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \cdots\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds x_{n - 1} x_n\) |
Proof
By definition:
| \(\ds e_2 \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right)\) | \(=\) | \(\ds \sum_{1 \mathop \le j_1 < j_2 \mathop \le n} \left({\prod_{i \mathop = 1}^2 x_{j_i} }\right)\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{1 \mathop \le j_1 < j_2 \mathop \le n} x_{j_1} x_{j_2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds x_1 x_2 + x_1 x_3 + \cdots + x_1 x_n\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds x_2 x_3 + \cdots + x_2 x_n\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \cdots\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds x_{n - 1} x_n\) |
$\blacksquare$