$\inline&space;&space;\displaystyle&space;f:A&space;\rightarrow&space;B$ This defines a function $\inline&space;f$ whose arguments take values from the set $\inline&space;A$ and which returns a unique value from the set $\inline&space;B$ for each corresponding argument. For example consider the function $\inline&space;f:\mathbb{R}&space;\rightarrow&space;\mathbb{R}_+$, given by $\inline&space;f(x)&space;=&space;10x$. By evaluating this function at various values of the argument the following results are obtained, $\inline&space;f(10.2)&space;=&space;102$, $\inline&space;f(-5.8)&space;=&space;-58$, $\inline&space;f(0)&space;=&space;0$. $\inline&space;&space;\displaystyle&space;f&space;\circ&space;g$ This defines the composite function $\inline&space;f&space;\circ&space;g$ defined by the relation $\inline&space;f&space;\circ&space;g&space;(x)&space;=&space;f(g(x))$ for all values of $\inline&space;x$, where $\inline&space;f$ and $\inline&space;g$ are apropriately chosen functions. $\inline&space;\displaystyle&space;|x|$ The absolute value function defined as: $|x|&space;=&space;\left\{&space;\begin{array}{rl}&space;x,&space;&&space;\text{for&space;}&space;x&space;\geq&space;0&space;\\&space;-x,&space;&&space;\text{for&space;}&space;x&space;<&space;0&space;\\&space;\end{array}&space;\right.$ For example $\inline&space;|1.23|&space;=&space;1.23$, $\inline&space;|0|&space;=&space;0$, $\inline&space;|-3.14|&space;=&space;3.14$. $\inline&space;\displaystyle&space;x&space;\approx&space;\alpha$ This tells that $\inline&space;x$ has an approximate value of $\inline&space;\alpha$, in other words $\inline&space;|x&space;-&space;\alpha|&space;<&space;\varepsilon$ where $\inline&space;\varepsilon$ is a very small positive value. For example $\inline&space;\pi&space;\approx&space;3.1415926535898$. $\inline&space;\displaystyle&space;x&space;\gg&space;\alpha$ This tells that $\inline&space;x$ has a much greater value than $\inline&space;\alpha$, for example $\inline&space;10000&space;\gg&space;5.34$. $\inline&space;&space;\displaystyle&space;x&space;\ll&space;\alpha$ This tells that $\inline&space;x$ has a much smaller value than $\inline&space;\alpha$, for example $\inline&space;-31000&space;\ll&space;0$. $\inline&space;\displaystyle&space;{\rm&space;sign}(x)$ The signum function defined as: $\rm&space;sign(x)&space;=&space;\left\{&space;\begin{array}{rl}&space;1,&space;&&space;\text{for&space;}&space;x&space;>&space;0&space;\\&space;0,&space;&&space;\text{for&space;}&space;x&space;=&space;0&space;\\&space;-1,&space;&&space;\text{for&space;}&space;x&space;<&space;0&space;\end{array}&space;\right&space;.$ For example $\inline&space;{\rm&space;sign}(-1034.5)&space;=&space;-1$, $\inline&space;{\rm&space;sign}(0)&space;=&space;0$, $\inline&space;{\rm&space;sign}(11.3)&space;=&space;1$. $\inline&space;&space;\displaystyle&space;\lfloor&space;x&space;\rfloor$ The floor function which gives the largest integer less than or equal to $\inline&space;x$. For example $\inline&space;\lfloor&space;2.8&space;\rfloor&space;=&space;2$, $\inline&space;\lfloor&space;-3.3&space;\rfloor&space;=&space;-4$. $\inline&space;&space;\displaystyle&space;\lceil&space;x&space;\rceil$ The ceiling function which gives the smallest integer not less than $\inline&space;x$. For example $\inline&space;\lceil&space;5.4&space;\rceil&space;=&space;6$, $\inline&space;\lceil&space;-2.8&space;\rceil&space;=&space;-2$. $\inline&space;\displaystyle&space;\sqrt{x}$ The square root of $\inline&space;x$ is a non-negative real number $\inline&space;y$ such that $\inline&space;y^2&space;=&space;x$. Obviously $\inline&space;x$ always has to be non-negative. For example $\inline&space;\sqrt{144}&space;=&space;12$, $\inline&space;\sqrt{256}&space;=&space;16$ while $\inline&space;\sqrt{-16}$ is not a proper expression since $\inline&space;-16$ is negative. $\inline&space;&space;\displaystyle&space;\sqrt[n]{x}$ The $\inline&space;n$-th root of $\inline&space;x$ is a real number $\inline&space;y$ such that $\inline&space;y^n&space;=&space;x$. If $\inline&space;n$ is even $\inline&space;x$ needs to be non-negative, as is the case for the square root with $\inline&space;n&space;=&space;2$. For example $\inline&space;\sqrt[3]{-8}&space;=&space;-2$, $\inline&space;\sqrt[5]{1024}&space;=&space;4$, while $\inline&space;\sqrt[4]{-256}$ is not a proper expression since the order of the root $\inline&space;n$ is even but $\inline&space;-256$ is negative. $\inline&space;&space;\displaystyle&space;\sum_{i=1}^n&space;E(i)$ If $\inline&space;E(i)$ is an expression that depends on the value of $\inline&space;i$, this evaluates the following sum $\sum_{i=1}^n&space;E(i)&space;=&space;E(1)&space;+&space;E(2)&space;+&space;E(3)&space;+&space;\ldots&space;+&space;E(n).$ For example if $\inline&space;E(i)&space;=&space;\sin(i\alpha)$ then the above sum becomes $\sum_{i=1}^n&space;\sin(i\alpha)&space;=&space;\sin(\alpha)&space;+&space;\sin(2\alpha)&space;+&space;\sin(3\alpha)&space;+&space;\ldots&space;+&space;\sin(n\alpha).$ $\inline&space;&space;\displaystyle&space;\sum_{P(i)}&space;E(i)$ If $\inline&space;E(i)$ is an expression that depends on the value of $\inline&space;i$, this evaluates the sum over those indices $\inline&space;i$ for which the predicate $\inline&space;P(i)$ becomes true. For example if $\inline&space;E(i)&space;=&space;\sin(i)$ and the predicate is $\inline&space;P(i)$: "$\inline&space;i$ is a prime and is less than $\inline&space;12$" then the sum becomes $\sum_{\displaystyle&space;i&space;\mbox{&space;is&space;a&space;prime}&space;\atop{\displaystyle&space;i&space;<&space;12}}&space;\sin(i)&space;=&space;&space;&space;\sin(2)&space;+&space;\sin(3)&space;+&space;\sin(5)&space;+&space;\sin(7)&space;+&space;\sin(11).$ Notice that if the predicate consists of several conditions, these are written one below the other as above. By letting $\inline&space;P(i)$: "$\inline&space;1&space;\leq&space;i&space;\leq&space;n$" this generalised sum becomes the previous easier sum $\sum_{1&space;\leq&space;i&space;\leq&space;n}&space;E(i)&space;=&space;\sum_{i=1}^n&space;E(i).$ $\inline&space;&space;\displaystyle&space;\prod_{i=1}^n&space;E(i)$ If $\inline&space;E(i)$ is an expression that depends on the value of $\inline&space;i$, this evaluates the following product $\prod_{i=1}^n&space;E(i)&space;=&space;E(1)&space;\cdot&space;E(2)&space;\cdot&space;E(3)&space;\cdot&space;\ldots&space;\cdot&space;E(n).$ For example if $\inline&space;E(i)&space;=&space;i&space;+&space;\sqrt{2&space;\cdot&space;i}$ then the above product becomes $\prod_{i=1}^n&space;i&space;+&space;\sqrt{2&space;\cdot&space;i}&space;=&space;(1&space;+&space;\sqrt{2&space;\cdot&space;1})&space;\cdot&space;(2&space;+&space;\sqrt{2&space;\cdot&space;2})&space;\cdot&space;(3&space;+&space;\sqrt{2&space;\cdot&space;3})&space;\cdot&space;\ldots&space;\cdot&space;(n&space;+&space;\sqrt{2&space;\cdot&space;n}).$ $\inline&space;&space;\displaystyle&space;\prod_{P(i)}&space;E(i)$ If $\inline&space;E(i)$ is an expression that depends on the value of $\inline&space;i$, this evaluates the product over those indices $\inline&space;i$ for which the predicate $\inline&space;P(i)$ becomes true. For example if $\inline&space;E(i)&space;=&space;i$ and the predicate is $\inline&space;P(i)$: "$\inline&space;i$ is a prime and is less than $\inline&space;16$" then the product becomes $\sum_{\displaystyle&space;i&space;\mbox{&space;is&space;a&space;prime}&space;\atop{\displaystyle&space;i&space;<&space;16}}&space;i&space;=&space;&space;&space;2&space;\cdot&space;3&space;\cdot&space;5&space;\cdot&space;7&space;\cdot&space;11&space;\cdot&space;13.$(1) Notice that if the predicate consists of several conditions, these are written one below the other as above. By letting $\inline&space;P(i)$: "$\inline&space;1&space;\leq&space;i&space;\leq&space;n$" this generalised product becomes the previous easier product $\prod_{1&space;\leq&space;i&space;\leq&space;n}&space;E(i)&space;=&space;\prod_{i=1}^n&space;E(i).$(2) $\inline&space;&space;\displaystyle&space;n!$ The factorial function of argument $\inline&space;n&space;\in&space;\mathbb{N}$ defined through $n!&space;=&space;\prod_{i=1}^n&space;i&space;\qquad&space;0!&space;=&space;1.$ $\inline&space;&space;\displaystyle&space;\binom{n}{k}$ The binomial coefficient of the natural numbers $\inline&space;n$ and $\inline&space;k$ defined through $\binom{n}{k}&space;=&space;\frac{n!}{k!(n-k)!}&space;\qquad&space;n&space;\geq&space;k&space;\geq&space;0.$