Inequality (mathematics)
An inequality is a subject of mathematics can be formulated with the size comparisons and examined. Each inequality consists of two terms which are < ( less-than symbol ), ≤ ( Less than or equal sign) ≥ ( Greater than or equal sign ) or> ( greater than sign ) through a comparison of the characters.
And are two terms, then, for example, is an inequality. This is called " less ( than) ". As in an equation is the left side and the right side of the inequality.
The values appearing in the two terms are usually real numbers. The issue raised by the reference character order relation then refers to the natural arrangement of real numbers.
- 4.1 triangle inequality
- 4.2 Cauchy- Schwarz inequality
- 4.3 Further inequalities
- 5.1 Complex Numbers
- 5.2 column vectors
- 5.3 Further Examples
Forms of inequalities
The following five forms of inequalities are possible:
The mold (5) is produced by the negation of an equation. It is therefore not specifically addressed in mathematics generally.
Inequalities are forms of expression. The occurring on both sides of an inequality functional terms usually include variables that are representative of elements from the domain of the respective terms. If these variables are fixed by elements of the respective domains replaced ( onset ), so arise statements which are either true or false.
Transformation of inequalities
Similar equations it is possible to transform these into equivalent inequalities even when inequalities.
Equivalent inequalities have the same solution set, so the forming of inequalities is important for solving inequalities, what will go down thereto the following section.
The following laws are shown for comparison
Reversibility
Inequalities can be reversed:
Monotony laws
Addition and subtraction
For arbitrary terms, and the following applies:
- It is precisely when.
- It is precisely when.
It may therefore be subtracted without changing the solution set of the inequality to both sides of an inequality adds the same terms or. For example, the inequality simplified by subtraction of the term on both sides of the inequality equivalent.
Multiplication and division
For arbitrary terms, and the following applies:
- It follows from.
- It follows from.
- It follows from and and.
- Is off and, then and.
Here, therefore, applies the following rule of thumb:
Thus, for example, the equivalent inequalities and how with the help of division by looks.
Applying a function
By applying a strictly monotonic function on both sides of the inequality we obtain an inequality with the same solution set as the Ausgangsungleichung. Similar to the monotony laws under certain circumstances the comparison character must be rotated. Applying on both a strictly increasing function, so the comparison does not change sign. Applying the other hand, is a strictly decreasing function, so the sign must be replaced < by the corresponding inverse characters >. The same also applies to the comparison character ≤ and ≥.
The natural logarithm and the square root function are strictly increasing functions and can therefore be used for transforming equations. Be two terms is then
Solving inequalities
A question when dealing with inequalities is - similar to the solution of equations - the question of the solution set of the inequality. Here is to answer the question whether, and if so deliver what elements of the domains when inserted into the two terms a true or false statement. An important technique for finding the solution set is the reshaping of inequality in a simpler form.
Known inequalities
In all mathematical sciences, there are records for the validity of inequalities. This means that certain mathematical statements secure under certain circumstances, the accuracy of a given inequality for a certain set of definitions. The following are some important inequalities are briefly mentioned.
Triangle inequality
After the triangle is a triangle the sum of the lengths of two sides and always at least as large as the length of the third side. That is formal.
This inequality can be generalized for many mathematical objects. For example, the following inequality
For the absolute value function is a generalization of the aforementioned inequality and applies to all real numbers. It also bears the name of triangle inequality. This inequality can be generalized to complex numbers or amount for integrals ( see Minkowski 's inequality ).
Cauchy- Schwarz inequality
Be pre-Hilbert space that is a vector space with scalar product and be and elements, then always the inequality
Equality holds if and only if and are linearly dependent. Vector spaces with scalar occur in many mathematical sciences. Therefore, the Cauchy-Schwarz inequality in many sub-disciplines of mathematics is important, for example, it is used in linear algebra, integration theory and probability theory.
Other inequalities
- Bernoulli's inequality
- Bonferroni inequalities
- Jensen's inequality
- Markov 's inequality
- Central inequality
- Chebyshev
- Different inequalities, which are named after Young
Extension of the term
Until now, only inequalities were considered in this article, the terms take values in the real numbers. The Ungleichungsbegriff is sometimes - but not uniform - for example, extended to complex numbers, vectors or matrices. In order to view inequalities for these objects must first remove the four comparison symbols <, ≤, > and ≥ - hereinafter also referred Relations - be defined for these objects.
Complex Numbers
The set of complex numbers, together with the usual addition and multiplication one body, but it is not possible to select a ratio ≤ a way that becomes a parent body. That is, it is not possible to meet a relation of both the trichotomy, the transitivity and monotony law. Sometimes, however, a relation represented by
Is defined, as viewed. Here denote complex numbers and the real part or the imaginary part of a complex number. This definition of the relation satisfies the trichotomy and the Transitivitätsgesetz.
Column vectors
For column vectors, it is possible to define relations. Be prepared with two column vectors and where and are real numbers. Relations on you can then, for example by
And by
Define. Analogously, one can explain the relations ≥ and >. Here, it is not possible to compare all the elements together. For example, none of the four comparison sign a relationship between the elements and describe.
Other examples
- If, as you defined if and only if is positive definite. If so exactly true when. Similarly, or ( semi-definite ) can be defined.
- Be a real Banach space and a cone. If so exactly true when.