basic

Basic notations

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Basic

$\inline \displaystyle f:A \rightarrow B$			This defines a function $\inline f$ whose arguments take values from the set $\inline A$ and which returns a unique value from the set $\inline B$ for each corresponding argument. For example consider the function $\inline f:\mathbb{R} \rightarrow \mathbb{R}_+$ , given by $\inline f(x) = 10x$ . By evaluating this function at various values of the argument the following results are obtained, $\inline f(10.2) = 102$ , $\inline f(-5.8) = -58$ , $\inline f(0) = 0$ .



$\inline \displaystyle f \circ g$			This defines the composite function $\inline f \circ g$ defined by the relation $\inline f \circ g (x) = f(g(x))$ for all values of $\inline x$ , where $\inline f$ and $\inline g$ are apropriately chosen functions.



$\inline \displaystyle \|x\|$			The absolute value function defined as: $\|x\| = \left\{ \begin{array}{rl} x, & \text{for } x \geq 0 \\ -x, & \text{for } x < 0 \\ \end{array} \right.$ For example $\inline \|1.23\| = 1.23$ , $\inline \|0\| = 0$ , $\inline \|-3.14\| = 3.14$ .



$\inline \displaystyle x \approx \alpha$			This tells that $\inline x$ has an approximate value of $\inline \alpha$ , in other words $\inline \|x - \alpha\| < \varepsilon$ where $\inline \varepsilon$ is a very small positive value. For example $\inline \pi \approx 3.1415926535898$ .



$\inline \displaystyle x \gg \alpha$			This tells that $\inline x$ has a much greater value than $\inline \alpha$ , for example $\inline 10000 \gg 5.34$ .



$\inline \displaystyle x \ll \alpha$			This tells that $\inline x$ has a much smaller value than $\inline \alpha$ , for example $\inline -31000 \ll 0$ .



$\inline \displaystyle {\rm sign}(x)$			The signum function defined as: $\rm sign(x) = \left\{ \begin{array}{rl} 1, & \text{for } x > 0 \\ 0, & \text{for } x = 0 \\ -1, & \text{for } x < 0 \end{array} \right .$ For example $\inline {\rm sign}(-1034.5) = -1$ , $\inline {\rm sign}(0) = 0$ , $\inline {\rm sign}(11.3) = 1$ .



$\inline \displaystyle \lfloor x \rfloor$			The floor function which gives the largest integer less than or equal to $\inline x$ . For example $\inline \lfloor 2.8 \rfloor = 2$ , $\inline \lfloor -3.3 \rfloor = -4$ .



$\inline \displaystyle \lceil x \rceil$			The ceiling function which gives the smallest integer not less than $\inline x$ . For example $\inline \lceil 5.4 \rceil = 6$ , $\inline \lceil -2.8 \rceil = -2$ .



$\inline \displaystyle \sqrt{x}$			The square root of $\inline x$ is a non-negative real number $\inline y$ such that $\inline y^2 = x$ . Obviously $\inline x$ always has to be non-negative. For example $\inline \sqrt{144} = 12$ , $\inline \sqrt{256} = 16$ while $\inline \sqrt{-16}$ is not a proper expression since $\inline -16$ is negative.



$\inline \displaystyle \sqrt[n]{x}$			The $\inline n$ -th root of $\inline x$ is a real number $\inline y$ such that $\inline y^n = x$ . If $\inline n$ is even $\inline x$ needs to be non-negative, as is the case for the square root with $\inline n = 2$ . For example $\inline \sqrt[3]{-8} = -2$ , $\inline \sqrt[5]{1024} = 4$ , while $\inline \sqrt[4]{-256}$ is not a proper expression since the order of the root $\inline n$ is even but $\inline -256$ is negative.



$\inline \displaystyle \sum_{i=1}^n E(i)$			If $\inline E(i)$ is an expression that depends on the value of $\inline i$ , this evaluates the following sum $\sum_{i=1}^n E(i) = E(1) + E(2) + E(3) + \ldots + E(n).$ For example if $\inline E(i) = \sin(i\alpha)$ then the above sum becomes $\sum_{i=1}^n \sin(i\alpha) = \sin(\alpha) + \sin(2\alpha) + \sin(3\alpha) + \ldots + \sin(n\alpha).$



$\inline \displaystyle \sum_{P(i)} E(i)$			If $\inline E(i)$ is an expression that depends on the value of $\inline i$ , this evaluates the sum over those indices $\inline i$ for which the predicate $\inline P(i)$ becomes true. For example if $\inline E(i) = \sin(i)$ and the predicate is $\inline P(i)$ : " $\inline i$ is a prime and is less than $\inline 12$ " then the sum becomes $\sum_{\displaystyle i \mbox{ is a prime} \atop{\displaystyle i < 12}} \sin(i) = \sin(2) + \sin(3) + \sin(5) + \sin(7) + \sin(11).$ Notice that if the predicate consists of several conditions, these are written one below the other as above. By letting $\inline P(i)$ : " $\inline 1 \leq i \leq n$ " this generalised sum becomes the previous easier sum $\sum_{1 \leq i \leq n} E(i) = \sum_{i=1}^n E(i).$



$\inline \displaystyle \prod_{i=1}^n E(i)$			If $\inline E(i)$ is an expression that depends on the value of $\inline i$ , this evaluates the following product $\prod_{i=1}^n E(i) = E(1) \cdot E(2) \cdot E(3) \cdot \ldots \cdot E(n).$ For example if $\inline E(i) = i + \sqrt{2 \cdot i}$ then the above product becomes $\prod_{i=1}^n i + \sqrt{2 \cdot i} = (1 + \sqrt{2 \cdot 1}) \cdot (2 + \sqrt{2 \cdot 2}) \cdot (3 + \sqrt{2 \cdot 3}) \cdot \ldots \cdot (n + \sqrt{2 \cdot n}).$



$\inline \displaystyle \prod_{P(i)} E(i)$			If $\inline E(i)$ is an expression that depends on the value of $\inline i$ , this evaluates the product over those indices $\inline i$ for which the predicate $\inline P(i)$ becomes true. For example if $\inline E(i) = i$ and the predicate is $\inline P(i)$ : " $\inline i$ is a prime and is less than $\inline 16$ " then the product becomes $\sum_{\displaystyle i \mbox{ is a prime} \atop{\displaystyle i < 16}} i = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13.$ (2) Notice that if the predicate consists of several conditions, these are written one below the other as above. By letting $\inline P(i)$ : " $\inline 1 \leq i \leq n$ " this generalised product becomes the previous easier product $\prod_{1 \leq i \leq n} E(i) = \prod_{i=1}^n E(i).$ (3)



$\inline \displaystyle n!$			The factorial function of argument $\inline n \in \mathbb{N}$ defined through $n! = \prod_{i=1}^n i \qquad 0! = 1.$



$\inline \displaystyle \binom{n}{k}$			The binomial coefficient of the natural numbers $\inline n$ and $\inline k$ defined through $\binom{n}{k} = \frac{n!}{k!(n-k)!} \qquad n \geq k \geq 0.$