Mathematical notation

Mathematical notations is kind of prefessional slang used mainly among researchers and, especially, among mathematicians. These notaitons are supposed to be already known, and usually the definition of these notationst are not repeated in scientific texts. For example, letter i is reserved for the imaginary unit; name exp  is reserved for the esponentiation, and letter $$\pi$$ is reserved for the period of the exponentiation, divided by 2i. In similar way, the character $$\mathbb{\emptyset}$$ denotes the empty set (which has no elements at all), $$\mathbb{N}$$ denotes the set of integer numbers, $$\mathbb{R}$$ denotes the set of real numbers, $$\mathbb{C}$$ denotes the set of complex numbers, and so on.

Overview
The basic concepts of this slang include grouping, that allows to combine several objects in one. Usually, the grouping is denoted with parenthesis. Also, the parenthesis are used to incifate the argument of operations; especially, if some operations $$A,B,C$$ from some set (called group) can be applied sequentially, one by one, in raw, for example, $$A\Big(B\big(C(z)\big)\Big)$$.

All things mathematicians deal with are called objects, and each object is supposed to belong to some set of objects, which is either already defined, or allows some independent definition. The possibility of such independent definition is especially important in order to exclude from the consideration such things as set of all possible sets which easy lead to paradoxes, at very beginning.

Mathematicians like to give names to all objects they deal with. Some ot these names are so established, that they are supposed to known a priori, for example, the equality, basic arithmetical operations, natural numbers, numbers e and $$\pi$$, etvetera. Such names form the basics of the mathematical notations. Variety of such mathematical notations is numerous, but mathematicians (and especially, editors) apply certain efforts in order to keep some standard of mathematical notations. For ecample, if some author needs to use characters e or $$\pi$$ or $$\sum$$ or $$\prod$$ in some non-usual way, say, as variables, then this author has to present serious reasons for such a strange choice of names of variables.

Equvalence $$=$$
Equivalence is renoted with symbol $$=$$. this symbol means that in some specific consideration, mathematicians makes no need to distinguish objects that appear at the left hand side of this character from the object that appear in the hight hand side. In the most of cases, the equvalence is used to compare numbers.

Exceptions
This rool has an exception. Sometimes, one may say "the integrand in the right hand side of the equation (3,14) becomes zero at..." However, in this case, one means not the mathematical object, but the way it is expressed in specific formila, typed in a book or in a paper.

True and False
In the simplest case, the mathematical exression may have only one of two values, true or false. These are basic mathematical expressions and basic mathematical notations. By default, are expressions in a scientific text are supposed to have value true. In some programming lamguages, at initialization, all the logical variables are set to false. For variables that may have only one of these two values, operations of boolean algebra are defined: and, or, not' for these operations, the following notations are in use:

Sets
In order to express operations with sets, the following symbols are used:
 * $$=$$ which means that the sets are [[equal
 * $$\subset$$ which subset

Quantors

 * $$\forall$$ for all
 * $$\exist$$

Nubers
Basic numbers are 0 and 1. With respect to these numbers, the arithmetic operations are defined. They are denoted with
 * $$++$$ (which has ony one argument)
 * $$\exp$$
 * $$\exp$$
 * $$\exp$$

and so on. At least for integer, all next operation in this raw can be constructed as recurrence of operations from previous rows. Most of conventional calculus is based on the operations 1-3 above. Some of numbers have special single-character notations: 0,1,2,3,4,5,6,7,8,9. For example,
 * $$ 1=0++$$
 * $$ 2=1++$$

and so on. Most of larger numbers have no special manes; the integer numbers are denoted using the positional numeral system. Inverse operations (if exist) are denoted, correspondently, with symbols
 * $$\log$$ and $$~^*\sqrt{~}$$
 * $$\log$$ and $$~^*\sqrt{~}$$
 * $$\log$$ and $$~^*\sqrt{~}$$
 * $$\log$$ and $$~^*\sqrt{~}$$

and so on.