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]
- (minor)
- (less than or equal)
- (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
It is
, who are read ”
And greater or equal a
” It is ”
And less than or equal ad
“.
The inequality in the strict sense indicates the equivalent scriptures instead
, light ”
And greater Of
” It is ”
And lesser Of
“.
This notation can be confused with the graphically similar notation
(O
), used with two different meanings: both to indicate that a number is sufficiently bigger than another (”
it is much greater than
“), both to indicate that a function is asypistically larger than another (”
Interested
“). 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
It is
distinct from each other, it always appears that
It is in connection with
, or that
It is in connection with
[2] .
A non -total relationship is called partial order report.
For example as a whole
the relationship ”
“It is total because it is possible to compare all the elements of the whole. If instead the report is considered together”
multiple of
“, this is a partial relationship because for example
It is not a multiple of
.
Antisimmetry and trichotomy [ change | Modifica Wikitesto ]
If inequality is narrow, then the property of trichotomy is worth:
- is worth one and only one of the three relationships .
If inequality is wide, then the antisymmetry is worth:
- .
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
It is
it is equivalent to checking whether their difference
it is positive or negative, or to compare
It is
. Furthermore
, as well as
.
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 It is ,
The same applies to inequality in a wide sense.
For the previous property, the second line is equivalent to the first, writing
in the place of
.
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
It is a function at real values. With this notation we mean that
It assumes only strictly positive values, namely that
in the domain of
. In this case there is a sign of a function or, equivalently, of the set of positivity of a function. Alike,
in the common domain of
It is
. 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. p.568
- ^ Massimo Bergamini, Graziella Barozzi, Anna Trifone, Mathematics.blu (second edition) Vol.1 , Zanichelli – Bologna, 2018, isbN 978-88-08-220-2208-1. p.236
- ^ Marzia Re Fraschini, Gabriella Grazzi, The principles of mathematics (volume 1) , Atlas, 2012, ISBN 978-88-268-1680-7. p.140
- 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.
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