Inequality – Wikipedia

In mathematics one inequality (O inequality ) is a total order relationship on the set of real numbers or on its subset, that is, it establishes a relationship between the numbers using the symbols of inequality, which are: ^[first]

• 
  ${Displaystyle <}$
  after-content-x4

  (minor)

• 
  ${Displaystyle>}$

  
  ${displaystyle leq }$

  (less than or equal)

• 
  ${displaystyle geq }$

  (greater or equal)

The first two express inequality in the sense strict , the last two express inequality in the sense long .

The same symbols can be used to “compare” two functions to real values.

The inequality in the wide sense is indicated with the equivalent scriptures

${displaystyle ageqslant b}$

It is

${displaystyle bleqslant a}$

, who are read ”

${displaystyle a}$

And greater or equal a

${displaystyle b}$

” It is ”

${displaystyle b}$

And less than or equal ad

${displaystyle a}$

“.

The inequality in the strict sense indicates the equivalent scriptures instead

${displaystyle a>b}$

${displaystyle b$

, light ”

${displaystyle a}$

And greater Of

${displaystyle b}$

” It is ”

${displaystyle b}$

And lesser Of

${displaystyle a}$

“.

This notation can be confused with the graphically similar notation

${displaystyle agg b}$

(O

${displaystyle bll a}$

), used with two different meanings: both to indicate that a number is sufficiently bigger than another (”

${displaystyle a}$

it is much greater than

${displaystyle b}$

“), both to indicate that a function is asypistically larger than another (”

${displaystyle a}$

Interested

${displaystyle b}$

“). In both cases it is not an inequality, but only a partial order relationship, that is, it may not allow to compare two distinct elements of the whole.

Total order [ change | Modifica Wikitesto ]

A order (broad or narrow) order) defined in a whole is total If, considering any elements of the set two

${displaystyle a}$

It is

${displaystyle b}$

distinct from each other, it always appears that

${displaystyle a}$

It is in connection with

${displaystyle b}$

, or that

${displaystyle b}$

It is in connection with

${displaystyle a}$

^[2] .

A non -total relationship is called partial order report.

For example as a whole

${displaystyle I=left{2,3,6,9,18right}}$

the relationship ”

${displaystyle a$

“It is total because it is possible to compare all the elements of the whole. If instead the report is considered together”

${displaystyle a}$

multiple of

${displaystyle b}$

“, this is a partial relationship because for example

${Displaystyle 9}$

It is not a multiple of

${Displaystyle 2}$

.

Antisimmetry and trichotomy [ change | Modifica Wikitesto ]

If inequality is narrow, then the property of trichotomy is worth:

${displaystyle forall a,b}$

is worth one and only one of the three relationships

${displaystyle a>b,;;;a$

.

If inequality is wide, then the antisymmetry is worth:

${displaystyle forall a,bqquad aleqslant bquad {text{e}}quad bleqslant aimplies a=b}$

.

Sum and subtraction [ change | Modifica Wikitesto ]

The inequalities are preserved if both terms are added or subtracted the same number ^[3] :

The same applies to inequality in a wide sense.

This property indicates that comparing two numbers

${displaystyle a}$

It is

${displaystyle b}$

it is equivalent to checking whether their difference

${displaystyle a-b}$

it is positive or negative, or to compare

${displaystyle a-b}$

It is

${Displaystyle 0}$

. Furthermore

${displaystyle a>0}$

${displaystyle -a<0}$

, as well as

${displaystyle a>b}$

${displaystyle -a<-b}$

.

This property in general describes the ordered groups.

Multiplication and division [ change | Modifica Wikitesto ]

The inequalities are preserved if both terms are multiplied or divided by the same strictly positive number. Multiplying or dividing by a strictly negative number, however, the inequalities exchange:

• for each triad of real numbers
  ${displaystyle a,b}$

  It is

  ${displaystyle c}$

  ,

The same applies to inequality in a wide sense.

For the previous property, the second line is equivalent to the first, writing

${displaystyle -c}$

in the place of

${displaystyle c}$

.

These properties in general describe ordered rings and ordered fields (or real fields).

Monotonal functions [ change | Modifica Wikitesto ]

The inequalities are the basis of the definition of the monotòne functions: the functions that retain or reversing the system of real numbers, therefore inequalities, are monotonous functions growing O decreasing .
In particular, the monotonal functions in the strict sense “maintain” inequalities in the strict sense; Instead a monotonous function in the wide sense provides only inequalities in the wide sense.

Sometimes it is abused of the notation for inequality, writing

${displaystyle f>0}$

${displaystyle f}$

It is a function at real values. With this notation we mean that

${displaystyle f}$

It assumes only strictly positive values, namely that

${displaystyle f(x)>0}$

${displaystyle x}$

in the domain of

${displaystyle f}$

. In this case there is a sign of a function or, equivalently, of the set of positivity of a function. Alike,

${displaystyle f>g}$

${displaystyle f-g>0}$

${displaystyle f(x)>g(x)}$

${displaystyle x}$

in the common domain of

${displaystyle f}$

It is

${displaystyle g}$

. The same happens with inequality in the wide sense.
When the domain of the functions is not specified, we speak of a snack.

Some “famous” inquaginances in mathematics are listed below.

• Massimo Bergamini, Graziella Barozzi, Anna Trifone, Mathematics.blu (second edition) Vol.1 , Zanichelli – Bologna, 2018, isbN 978-88-08-220-2208-1.
• Marzia Re Fraschini, Gabriella Grazzi, The principles of mathematics (volume 1) , Atlas, 2012, ISBN 978-88-268-1680-7.