Math operation)

A mathematical operation is a function on a tuple that obtains a result, applying pre-established rules on the tuple.

1. A mathematical operation, to be considered as such, always has to guarantee a result, operations that for certain tuple values do not guarantee a result cannot be considered mathematical operations properly.

2. A mathematical operation has to give a single result, if for a given tuple it can present more than one result, it cannot be considered mathematical operation itself.

An important characteristic of a mathematical operation is the number of terminators of the tuple: arity. Being that of two terms: binary operation of great importance.

In algebra, the operations addition, subtraction, multiplication, and division are used. An operation is the application of an operator on the elements of a set that it contains. The operator takes the initial elements and relates them to another element of a final set that may or may not be of the same nature; this is technically known as the law of composition.

In arithmetic and calculus, the starting set can be formed by elements of a single type (arithmetic operations act only on numbers) or several (the product of a vector by a scalar encompasses the set union of vectors and scalars that make up a vector space.

Depending on how the sets involved in the operation are with respect to the set considered main according to our intentions, we can classify operations into two types: internal and external.

Operation properties

• The addendum operation (+)
  • written a+b{displaystyle ,a+b}
  • is switching: a+b=b+a{displaystyle ,a+b=b+a}
  • is associative: (a+b)+c=a+(b+c){displaystyle ,(a+b)+c=a+(b+c)}
  • has an inverse operation called subtraction: (a+b)− − b=a{displaystyle ,(a+b)-b=a}That's the same as adding the Opos, a− − b=a+(− − b){displaystyle ,a-b=a+(-b)}
  • has a neutral element 0 that does not alter the sum: a+0=a{displaystyle ,a+0=a}

• The multiplication operation (×)
  • written (a× × b){displaystyle ,(atimes b)} or (a⋅ ⋅ b){displaystyle ,(acdot b)}
  • is a repeat addition a× × n=a+a+...... +a{displaystyle atimes n=a+a+ldots +a} (n sometimes)
  • is switching: (a⋅ ⋅ b){displaystyle ,(acdot b)} = (b⋅ ⋅ a){displaystyle ,(bcdot a)}
  • is associative: (a⋅ ⋅ b)⋅ ⋅ c=a⋅ ⋅ (b⋅ ⋅ c){displaystyle ,(acdot b)cdot c=acdot (bcdot c}
  • abbreviated by juxtaposition: a⋅ ⋅ b≡ ≡ ab{displaystyle acdot bequiv ab}
  • has an inverse operation, for numbers different than zero, called division: (ab)b=a{displaystyle {frac {(ab)}{b}}=a}which is equal to multiply by reciprocal, ab=a(1b){displaystyle {frac {a}{b}}}=aleft({frac {1{b}}}}{b}}}}}{b}}}
  • has a neutral element 1 that does not alter multiplication: a× × 1=a{displaystyle atimes 1=a}
  • is distributive to the addendum: (a+b)⋅ ⋅ c=ac+bc{displaystyle ,(a+b)cdot c=ac+bc}

• The power operation
  • written ab{displaystyle ,a^{b}}
  • is a repeated multiplication: an=a× × a× × ...... × × a{displaystyle a^{n}=atimes atimes ldots times a} (n sometimes)
  • is neither comutative nor associative: in general abI was. I was. ba{displaystyle ,a^{b}neq b^{a} and (ab)cI was. I was. a(bc){displaystyle ,(a^{b})^{c}neq a^{(b^{c}}}}}}
  • has an inverse operation, called logarithmation: alorgab=b=lorgaab{displaystyle ,a^{log_{a}b}=b=log_{a}a^{b}}}
  • can be written in terms of root n-sima: am/n≡ ≡ (amn){displaystyle a^{m/n}equiv ({sqrt[{n}]{a^{m}}}}}}} and therefore the pair roots of negative numbers do not exist in the system of real numbers. (See: Complex Number System)
  • is distributive with respect to multiplication: (a⋅ ⋅ b)c=ac⋅ ⋅ bc{displaystyle ,(acdot b)^{c}=a^{c}cdot b^{c}}
  • has the property: ab⋅ ⋅ ac=ab+c{displaystyle {a^{b}}cdot {a^{c}}=a^{b+c}}}
  • has the property: (ab)c=abc{displaystyle ,(a^{b})^{c}=a^{bc}}

Order of operations

To complete the value of an expression, it is necessary to compute parts of it in a particular order, known as the order of precedence or the order of precedence of operations. First, the values of the expressions enclosed in group signs (parentheses, square brackets, braces) are calculated, then those of exponentiations, then multiplications and divisions, and finally, additions and subtractions.

Properties of equality

The relation of equality (=) is:

• Reflective: a=a{displaystyle ,a=a}
• symmetrical: yes a=b{displaystyle ,a=b} then. b=a{displaystyle ,b=a}
• transitive: yes a=b{displaystyle ,a=b} and b=c{displaystyle ,b=c} then. a=c{displaystyle ,a=c}

Laws of equality

The equality relation (=) has the following properties:

• Yeah. a=b{displaystyle ,a=b} and c=d{displaystyle ,c=d} then. a+c=b+d{displaystyle ,a+c=b+d} and ac=bd{displaystyle ,ac=bd}
• Yeah. a=b{displaystyle ,a=b} then. a+c=b+c{displaystyle ,a+c=b+c}
• If two symbols are equal, then one can be replaced by the other.
• regularity of the sum: working with real or complex numbers happens that if a+c=b+c{displaystyle ,a+c=b+c} then. a=b{displaystyle ,a=b}.
• conditional regularity of multiplication: if a⋅ ⋅ c=b⋅ ⋅ c{displaystyle ,acdot c=bcdot c} and c{displaystyle ,c} It's not zero, then a=b{displaystyle ,a=b}.

Laws of inequality

The inequality relation (<) has the following properties:

• of transitivity: yes <math alttext="{displaystyle ,aa.b{displaystyle , a cup}<img alt="{displaystyle ,a and <math alttext="{displaystyle ,bb.c{displaystyle ,b impliedc}<img alt="{displaystyle ,b then. <math alttext="{displaystyle ,aa.c{displaystyle , a impliedc}<img alt="{displaystyle ,a
• Yeah. <math alttext="{displaystyle ,aa.b{displaystyle , a cup}<img alt="{displaystyle ,a and <math alttext="{displaystyle ,cc.d{displaystyle ,c margind}<img alt="{displaystyle ,c then. <math alttext="{displaystyle ,a+ca+c.b+d{displaystyle ,a+c visb+d}<img alt="{displaystyle ,a+c
• Yeah. <math alttext="{displaystyle ,aa.b{displaystyle , a cup}<img alt="{displaystyle ,a and 0}" xmlns="http://www.w3.org/1998/Math/MathML">c▪0{displaystyle ,cpur0}0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d9b9ac0d3f580582cf3f0e9e8a12055501e413d" style="vertical-align: -0.338ex; width:5.655ex; height:2.176ex;"/> then. <math alttext="{displaystyle ,acac.bc{displaystyle ,acpensbc}<img alt="{displaystyle ,ac
• Yeah. <math alttext="{displaystyle ,aa.b{displaystyle , a cup}<img alt="{displaystyle ,a and <math alttext="{displaystyle ,cc.0{displaystyle ,c ingredient0}<img alt="{displaystyle ,c then. <math alttext="{displaystyle ,bcbc.ac{displaystyle ,bc visac}<img alt="{displaystyle ,bc

Rule of signs

In the product and in the quotient of positive (+) and negative (-) numbers the following rules are fulfilled:

{+⋅ ⋅ − − =− − +⋅ ⋅ +=+− − ⋅ ⋅ − − =+− − ⋅ ⋅ +=− − {displaystyle {begin{cases}+cdot -=-cdot +=+cdot -=+cdot +=-cdot +=-\end{cases}}}}}

Abstract Algebra

An operation f{displaystyle f_{}^{}{}}} is internal if both the initial and final elements belong to the only set A{displaystyle A_{}^{}{}}}.

f:AI→ → A,AI=A× × A× × I× × A= i한 한 IAi,I{displaystyle f:;A^{I}to A;,;A^{I}=Atimes Atimes cdots ^{I}times A=prod _{iin I}A_{i};, It's a set.

Which can also be expressed:

(a1,a2,a3, ,an)→fb{displaystyle (a_{1},a_{2},a_{3},cdotsa_{n});{xrightarrow {f}}{;b}

Or also:

f(a1,a2,a3, ,an)→ → b{displaystyle f(a_{1},a_{2},a_{3},cdotsa_{n});to ;b}

Depending on the nature of the initial Cartesian product of the operation we can differentiate:

• Finite operations if the initial set I{displaystyle I_{}^{}{}} is a fine cartesian product.

• Infinite operations Otherwise.

N-ary operation

We'll say that f{displaystyle f_{}^{}{}}} is a n-aria operation in the whole A{displaystyle A_{}^{}{}}}Yeah.

f:An→ → A{displaystyle f:A_{}{n}to A}

a n한 한 N{displaystyle n_{}{}{in mathbb {n} } it is called arbit or anity.

Binary operation

An operation is binary when n{displaystyle n} equals two:

⋆ ⋆ :A× × B→ → C(a,b)→ → c=a⋆ ⋆ b{displaystyle {begin{array}{rrcl}star: stranger;Atimes B blindfoldto " C alien(a,b) strangerto &c=astar bend{array}}}}}}

and also:

a⋆ ⋆ b→ → c{displaystyle astar b;to ;c}

(a,b)→⋆ ⋆ c{displaystyle(a,b);{xrightarrow {star }};c}

⋆ ⋆ (a,b)→ → c{displaystyle star (a,b);to ;c}

Example:

In the set of natural numbers, N{displaystyle mathbb {N} }the addition operation: +:N× × NΔ Δ N{displaystyle +:mathbb {N} times mathbb {N} longrightarrow mathbb {N} }, (N,+){displaystyle (N,+),}with the different expressions:

1. a,b,c한 한 N,a+b→ → c{displaystyle a,b,cin mathbb {N}quad a+bto c}
2. a,b,c한 한 N,(a,b)→+c{displaystyle a,b,cin mathbb {N}quad (a,b);{xrightarrow {+};c}
3. a,b,c한 한 N,+(a,b)→ → c{displaystyle a,b,cin mathbb {N}quad +(a,b);to ;c}

where a and b are the addends and c the result of the addition.

Unary operation

A unary operation, with a single parameter:

⋆ ⋆ :A→ → Ba→ → b=⋆ ⋆ (a){displaystyle {begin{array}{rrcl}star: stranger;A faketo > > > }

also often called functions.

Examples:

• Given the set of natural numbers N{displaystyle mathbb {N} }the operation would increase or follow:

in:N→ → Na→ → b=in(a){displaystyle {begin{array}{rrcl}in: stranger;N faketo &N\ strangera faketo &b=in(a)end{array}}}}}}}}

Where:

in(n)=n+1:n한 한 N{displaystyle in(n)=n+1;:;nin mathbb {N} }

• Given the whole numbers Z{displaystyle mathbb {Z} }the opposite operation, like:

orp:Z→ → Za→ → b=orp(a){displaystyle {begin{array}{rrcl}op: stranger;Z strangerto &Z strangera strangerto &b=op(a)end{array}}}}}}}

this is:

orp(e)=− − e:e한 한 Z{displaystyle op(e)=-e;:;ein mathbb {Z} }

0-ary operation

A 0-aria operation is when we have an operation. f:A0→ → A{displaystyle f:A^{0}to A} I mean:

⋆ ⋆ :{∅ ∅ !→ → A∅ ∅ → → b=⋆ ⋆ (∅ ∅ ){displaystyle {begin{array}{rrcl}star: alien;{emptyset }{to ' fakeemptyset ' as ' s (emptyset)end{array}}}}}

Example: A null operation usually returns constants, for example the value of pi:

pi:{∅ ∅ !→ → R∅ ∅ → → a=pi(∅ ∅ ){displaystyle {begin{array}{rcl}pi: stranger;{emptyset }{to > > > > > > > >

Which assigns a the real value of the number pi.

• An operation that designates a distinguished element of A{displaystyle A_{}^{}{}}}, in group theory would be the neutral element of a group.

External operation

A law of external composition on a set A with a set B is an application:

⋆ ⋆ :B× × A→ → A(a,b)→ → c=a⋆ ⋆ b{displaystyle {begin{array}{rrcl}star: stranger;Btimes A faketo > > bend{array}}}}

this application is said to be an external operation.

Example: Given the set V2{displaystyle V_{2};} of vectors in the plane and the set of scales R{displaystyle mathbb {R} } of real numbers, we have that the product of a real number by a vector in the plane is a vector in the plane:

⋅ ⋅ :R× × V2→ → V2(a,v→ → )→ → u→ → =a⋅ ⋅ v→ → {displaystyle {begin{array}{rrcl}cdot: fakertimes V_{2}{2}{2}{2}{2}{pos(a,{vec {v}}}}}}{cdot {cdot {vec}}{end{array}}}}}}

Given the vector:

v→ → =3i+6j{displaystyle {vec {v}}=3i+6j;}

If we multiply it by a scalar 3:

3⋅ ⋅ v→ → =3⋅ ⋅ (3i+6j)=(9i+18j)=u→ → {displaystyle 3cdot {vec {v}}=3cdot (3i+6j)=(9i+18j)={vec {u}}}}}

we can see that the two vectors are from the plane:

(3i+6j),(9i+18j)한 한 V2{displaystyle (3i+6j),(9i+18j)in V_{2}}}

Starting with distinct sets A and B, and an application:

⋆ ⋆ :A× × A→ → B(a,b)→ → c=a⋆ ⋆ b{displaystyle {begin{array}{rrcl}star: strangerAtimes A faketo > bend{array}}}}

it is also said to be a law of external composition. For example, the scalar product of two vectors in the plane, results in a real number, that is:

:V2× × V2→ → R(u→ → ,v→ → )→ → a=u→ → v→ → {displaystyle {begin{array}{rcl}circ: fakeV_{2}{2times V_{2}{2}{to &R nightmare({vec {u}}}},{vec {v}}}{vec}{{{vec}}}}}{end{array}}}}}}}}}}

Taking the vectors of the plane:

u→ → =(x1,and1){displaystyle {vec {u}}=(x_{1},y_{1}}}}

v→ → =(x2,and2){displaystyle {vec {v}}=(x_{2},y_{2}}}}

And being their scalar product:

u→ → v→ → =(x1,and1) (x2,and2)=x1⋅ ⋅ x2+and1⋅ ⋅ and2{displaystyle {vec {u}}}circ {vec {v}}=(x_{1},y_{1})circ (x_{2},y_{2})=x_{1}cdot x_{2}+y_{1}cdot y_{2}}}

That results in a real number, let's see a numerical example:

u→ → =(3,6){displaystyle {vec {u}}=(3.6)}

v→ → =(5,2){displaystyle {vec {v}}=(5,2)}

Operating

u→ → v→ → =(3,6) (5,2)=3⋅ ⋅ 5+6⋅ ⋅ 2=15+12=27{displaystyle {vec {u}}circ {vec {v}}=(3.6)circ (5.2)=3cdot 5+6cdot 2=15+12=27}