# Dashed line theory definitions and symbols

This page lists the various symbols which are used in our posts about dashed line theory. Some of them have been defined specifically for dashed line theory, while others are already known universally and are reported below just for reading convenience.

Symbol Meaning
$a \mathrm{\ mod\ } b$ Modulus: it indicates the remainder of the division between $a$ and $b$.

For example, $20 \mathrm{\ mod\ } 7 = 6$ indicates that the division between 20 and 7 gives 6 as remainder.

Modulus has priority over additions and subtractions, while multiplications, divisions and powers have priority over it. Then, given the following expression:

$a + (b + c) x \mathrm{\ mod\ } d^2 + e$

execution order of operations is:

$a + (((b + c) x) \mathrm{\ mod\ } (d^2)) + e$
$a \mathrm{\ mod^{\star}\ } b$ Modulus Star: it’s defined as

$a \mathrm{\ mod^{\star}\ } b := \begin{cases} b & \text{if }a \mathrm{\ mod\ } b = 0 \\ a \mathrm{\ mod\ } b & \text{otherwise} \end{cases}$

It behaves like $a \mathrm{\ mod\ } b$, but returns $b$ in place of 0. For example, if $b = 5$:

1 $\mathrm{mod}$ 5 = 1 $\mathrm{mod^{\star}}$ 5 = 1;
2 $\mathrm{mod}$ 5 = 2 $\mathrm{mod^{\star}}$ 5 = 2;
3 $\mathrm{mod}$ 5 = 3 $\mathrm{mod^{\star}}$ 5 = 3;
4 $\mathrm{mod}$ 5 = 4 $\mathrm{mod^{\star}}$ 5 = 4;
5 $\mathrm{mod}$ 5 = 0; 5 $\mathrm{mod^{\star}}$ 5 = 5.

Like modulus, it has priority only over additions and subtractions.

$(C)$ Integer value of a proposition: it’s defined as

$$(C) := \begin{cases} 1 &\text{if }C\text{ is true}\\ 0 &\text{if }C\text{ is false}\end{cases}$$

where $C$ is a proposition. For example, $(a \gt 10)$ is 1 if $a \gt 10$, 0 otherwise.

$\sigma_i(x_1, x_2, ... x_j)$ Elementary symmetric polynomials: they are defined as

$\sigma_0(x_1, x_2, x_3, x_4, \ldots, x_j) = 1$
$\sigma_1(x_1, x_2, x_3, x_4, \ldots, x_j) = x_1 + x_2 + x_3 + x_4 + \ldots + x_j$
$\sigma_2(x_1, x_2, x_3, x_4, \ldots, x_j) = x_1 \cdot x_2 + x_1 \cdot x_3 + x_1 \cdot x_4 + \ldots + x_2 \cdot x_3 + x_2 \cdot x_4 + \ldots + x_{j-1} \cdot x_j$
$\sigma_3(x_1, x_2, x_3, x_4, \ldots, x_j) = x_1 \cdot x_2 \cdot x_3 + x_1 \cdot x_2 \cdot x_4 + \ldots + x_2 \cdot x_3 \cdot x_4 + \ldots + x_{j-2} \cdot x_{j-1} \cdot x_j$
$\sigma_j(x_1, x_2, x_3, x_4, \ldots, x_j) = x_1 \cdot x_2 \cdot x_3 \cdot x_4 \cdot \ldots \cdot x_j$

Generally speaking, in order to compute $\sigma_i(x_1, x_2, ... x_j)$ you have to list all the combinations $i$ by $i$ if the $j$ arguments, transform each into a product, and then sum up the so obtained products: for example, $\sigma_2(3, 5, 7) = 3 \cdot 5 + 3 \cdot 7 + 5 \cdot 7$. If $i$ is zero, the arguments are irrelevant, since the result is always 1. Similarly, we also agree that the result is 1 also when the argument list is empty, i.e. $\sigma_i() = 1$ for all $i$.

$[a, b]$ Range: it’s the set of the integer numbers between $a$ and $b$: $[a,b] := \{n \in \mathbb{Z} \mid a \leq n \leq b\}$. The number $b - a + 1$, that is the range cardinality, is defined width of the range. It’s supposed that $a \leq b$.
$p, q$ The symbols $p$ and $q$ are used for indicating prime numbers, except if differently specified.
$p_i$ The symbol $p_i$ indicates the $i$-th prime number, with $i \geq 1$, so that $p_1 = 2$, $p_2 = 3$, $p_3 = 5$, etcetera.
$q_i$ The symbol $q_i$, with $i \geq 1$, indicates a number taken from a list of prime numbers, not necessarily sorted or complete. For example if the list is 5, 17, 3, then $q_1 := 5$, $q_2 := 17$ and $q_3 := 3$.