Mathematical notation

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Mathematics is supported by a formal symbolic language, mathematical notation, which follows a series of its own conventions. The symbols represent a concept, a relationship, an operation, or a mathematical formula according to certain rules. These symbols should not be considered abbreviations, but entities with their own and autonomous value.

Some basic principles are:

• The symbols of a letter are represented in italics: ${displaystyle scriptstyle a,,b,,i,,k,,x,,y}$etc.
• The symbols of several letters are represented in round lyrics: ${displaystyle scriptstyle cos alpha,exp x}$, etc.; instead of ${displaystyle scriptstyle ln x}$ should not be written ${displaystyle scriptstyle lnx}$, because that would represent the product ${displaystyle scriptstyle lcdot ncdot x}$ instead of the neperian logarithm.
• According to ISO/IEC 80,000 differential operators and constant universal maths (${displaystyle scriptstyle {text{e},,{text{i}}}}$), also written with round letter: ${displaystyle scriptstyle a{text{e}}{x}}$.

Set theory

Sean. ${displaystyle x}$ an element and ${displaystyle A,B}$ joints

Relationship	Notation	He reads
membership	${displaystyle xin A}$	x belongs to A
inclusion	${displaystyle Asubset B}$	A is contained in B
	${displaystyle Asubseq B}$	A is contained in B or equal to B
inclusion	${displaystyle Asupset B}$	A contains B
	${displaystyle Asupseteq B}$	A contains B or is equal to B

A crossbar on the symbol denies the statement; for example ${displaystyle xnot in A}$ It's "x" No. belongs to A";

Numeric sets

The following table lists some examples of symbols that use blackboard bold. The symbol created with LaTeX is shown, the equivalent Unicode character (might not be visible depending on the browser and available fonts), and its usual meaning in mathematics:

TeX	Unicode	Use in mathematics
${displaystyle mathbb {C} !}$	C	Complex numbers
${displaystyle mathbb {H} !}$	H	Cuaterniones
${displaystyle mathbb {N} !}$	N	Natural numbers
${displaystyle mathbb {P} !}$	P	Number of cousins
${displaystyle mathbb {Q} !}$	Q	Rational numbers
${displaystyle mathbb {R} !}$	R	Actual numbers
${displaystyle mathbb {S} !}$	S	Sphere
${displaystyle mathbb {Z} !}$	Z	Integer numbers

Special number sets

${displaystyle mathbb {N} ={1,2,3,ldots }}$

${displaystyle mathbb {N} _{0}=mathbb {N} ^{*}=mathbb {N} cup {0}={0,1,2,3,ldots }}}}$

${displaystyle mathbb {Z} ={ldots -3,-2,-1,0,1,2,3ldots }}$

${displaystyle mathbb {Z} ^{+}=mathbb {Z} setminus mathbb {Z} ^{-}={0,1,2,3,ldots }}}}}$

${displaystyle mathbb {Q} ={p:quad p={frac {a}{b}}}quad /quad a,bin mathbb {Z} quad land quad bneq 0}}}$

${displaystyle mathbb {Q} ={p:quad p={frac {a}{b}}}}}quad /quad ain mathbb {Z} quad land quad bin mathbb {N} }}}}}$

${displaystyle mathbb {R} ={mbox{The set of real numbers}}$

${displaystyle mathrm {Irrational} =mathbb {R} setminus mathbb {Q} }$

${displaystyle {overline {mathbb {R}}}=mathbb {R} cup {pm infty }={mbox{ The real straight enlarged$

${displaystyle mathbb {C} ={c:quad c=a+bcdot iquad /quad a,bin mathbb {R} quad land quad i^{2}=-1}}$

${displaystyle mathbb {s} ^{1}={zin mathbb {C} colon structurezin}$

Expressions

Relationship	Notation	He reads
equality	${displaystyle x=y}$	x is just like and
lower than	${displaystyle x visy}$	x is less than y
greater than	${displaystyle x 2005}$	x is greater than y
approach	${displaystyle xapprox and}$	x is approximately equal to and

Quantifier	Notation	He reads
universal quantification	${displaystyle forall x... !$	for all x
existential quantifier	${displaystyle exists x... !$	There is at least one x
existential quantifier with singleness mark	${displaystyle exists!x... !$	There is only one x
such as	${displaystyle xmid y}$	x, so and
therefore	${displaystyle xtherefore and}$	x, therefore and

Example:

Weierstrass's theorem:

"Let f be a continuous real function on a closed and bounded real interval [a, b], where a is strictly less than b.

You have to:

• The f function is set.
• The function f reaches a maximum and a minimum in that interval, not necessarily unique. "

This theorem can be expressed in mathematical notation as follows:

" ${displaystyle f:[a,b]subseq mathbb {R} longrightarrow mathbb {R}a habitbLongrightarrow exists r,sin [a,b]mid forall xin [a,b]:f(r)leq f(x)leq f(s)}$ ".

Propositional logic, Boolean algebra

Basic Operators

The most basic logical operators are conjunction, disjunction, and negation.

Sean. ${displaystyle p}$ and ${displaystyle q}$ two proposals

Operation	Notation	He reads
Negative	${displaystyle neg p}$	no 'p'
conjunction	${displaystyle pland q}$	'p' and 'q'
Disjunction	${displaystyle plor q}$	'p' or (exclusive) 'q'

The basic operators are used to form atomic declarations. The atomic statements tell which combination of pp and qq is true.

Implication

A very useful combination of mathematical operators is the implication. It's written ${displaystyle pto q}$ or ${displaystyle pRightarrow q}$ as abbreviation of ${displaystyle neg plor q}$. The Declaration "${displaystyle p}$ implies ${displaystyle q}$"it is always true that ${displaystyle p}$ it's true, but not necessarily if ${displaystyle q}$ it is (since it can be true for other reasons).

Yeah. ${displaystyle pRightarrow q}$ and ${displaystyle qRightarrow p}$, it is written ${displaystyle pLeftrightarrow q}$to read "${displaystyle p}$ implied and involved by ${displaystyle q}$"or"${displaystyle p}$ Yes and only if ${displaystyle q}$".

One of the most common uses of logical operators is in the Programming of Information Systems, as well as in the generation of electrical circuits, and in general in any decision-making system for the company or for daily life, For example:

If I leave home late and I don't have a car, then I'll be late for work.

conjunction: I'm late ${displaystyle land }$ I don't have a vehicle. ${displaystyle Rightarrow }$ I'll be late for work.

I travel by bus or I travel in my car, not both at the same time.

Logical disjunction: by bus ${displaystyle lor }$ I travel in my car ${displaystyle Rightarrow }$ or one or the other.

Contradictions of language

If we say: here there is nobody here and we literally apply the double negative expressed in our everyday speech, then, we could understand that there is someone here.

Logical denial: no ${displaystyle neg }$ There's nobody ${displaystyle Rightarrow }$ There's someone here.

If a company does not produce nothing, we could understand that the company produces something.

Logical denial: no ${displaystyle neg }$ produces nothing ${displaystyle Rightarrow }$ produces something.

Other languages, such as French, avoid this ambiguity or contradiction by delimiting the negation with a double mark, replacing only the second mark when using "nothing" or "no one", so when conjugating the negation only the second mark, "ne...pas" becomes "ne...rien" or "ne...personne", which avoids a possible double negative interpretation of the basic structure.

Quantifiers

So far the statements we can make do not say when they are true. To tell us when a statement is true, we need quantifiers. There are three basic quantifiers: the universal quantifier, the existential quantifier, and the uniquely marked existential quantifier. Here are the symbols.

Name	Notation	He reads
universal quantification	${displaystyle forall xldots }$	For all x...
existential quantifier	${displaystyle exists xldots }$	There is at least one x...
existential quantifier with singleness mark	${displaystyle exists}xldots }$	There's only one x...

Quantified statements are written in the form ${displaystyle forall x pquad oquad exists ymid} q$ to read "for everything" ${displaystyle x}$It's true that ${displaystyle p}$"and there is at least one ${displaystyle and}$ such as ${displaystyle q}$ It's true."

These last two quantifiers can be used for the same, as ${displaystyle neg forall x\ p}$ He says the same thing he says. ${displaystyle exists xintneg p}$. In words, saying "it's not for everything ${displaystyle x}$ that ${displaystyle p}$ It's true. It's just like saying "there is ${displaystyle x}$ such as ${displaystyle p}$ It's fake."

Number Theory

Mathematical analysis

Actual analysis

Limits

To say that the function limit ${displaystyle f}$ That's it. ${displaystyle L}$ When ${displaystyle x}$ tends to ${displaystyle a}$, it is written:

${displaystyle lim _{xto a}f(x)=L}$ or ${displaystyle f(x){overset {xto a}{rightarrow }}$ or simply ${displaystyle f(x)to L}$.

Likewise, to say that the succession ${displaystyle {a_{n}{n}}}$ Go. ${displaystyle a}$ When ${displaystyle n}$ tends to infinity, it is written:

${displaystyle lim _{nto infty }a_{n}=a}$ or ${displaystyle a_{n}to a}$.

Derivatives

Ordinary derivatives

The derivative of a function is defined as the limit of the quotient of the change in the ordinate and the abscissa. There are several notations to denote the derivative of a function of a single variable:

${displaystyle y=f(x),}$

The derivatives would be:

${displaystyle y'quad f'(x)quad {frac {d}{dx}f}(x)quad {frac}{dx}{dx}}quad D_{x}(y)quad D_{x}(f(x)}}}$

Partial derivatives

If the function depends on two or more variables, for example:

${displaystyle z=f(x,y),}$

The partial derivatives with respect to each of the independent variables:

${displaystyle {frac {partial z}{partial x}}quad {frac {partial}{partial x}}f(x,y)quad {frac {partial z}{partial y}}{quad {frac {partial }{partial y}}f(x,y}$

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