Well-defined

Mathematics can define an object not only by the defining equation (explicit), but also a characteristic of ( implicit). While an explicit definition is always permitted, an implicit definition is permitted only under the condition that it is actually exactly one object with the specified property. This condition is called the well- definedness of the implicit definition.

The proof of well- definedness can be decomposed into two parts: the existence and uniqueness of the concept to be defined.

Implicit definitions often emerge unnoticed if one defines illustrations on factor quantities. The result is defined by a mapping to a representative. Well- definedness of running out here on the independence of the result on the choice of representatives.

• 2.2.1 Definition of the induced link
• 2.2.2 Well- definedness of induced links
• 2.2.3 Examples of induced links

• 3.1 Definition of a function
• 3.2 Range of values ​​of a function
• 3.3 Linkages in groups
• 3.4 Well- definedness of quantities

Simple Example

To 1: The well- definedness of stating that there is exactly one number with the property for each number. This is in fact the case, since the squaring function from to is bijective. The function is thus well defined. is the square root function.

To 2: Well -definedness applies here not because true, for example, and. The uniqueness is violated.

To 3: Again, does not apply well- definedness, because has no real solutions. Existence is injured.

Representatives independence

In the literature, there is often the definition of well -definedness as representatives independence. Chance is expressly pointed out that there is no meaning beyond that.

The representatives independence should first be explained by an example. Every rational number can be as a fraction of two integers, the numerator and the denominator write. ' Define ' So, we, as a function, which assigns to any rational count her counter.

It is known, that is true, a contradiction! The definition of words can not be in order. The definition of is actually an implicit definition and it is not well defined. Let us look more closely at the definition of: the fraction stands for the equivalence class of all couples, applies. The definition of accurate would have to be: For all rational numbers is defined as the value for the there is a with. Well it turns out that there are several such - for these are, for example 1 (with ) or 2 ( with ). The well- definedness therefore does not apply.

For two mathematical concepts representatives independence must be demonstrated:

Induced pictures

Definition of the induced mapping

Given up and up two sets and as well as the equivalence relations. denote the equivalence class of the element relative and the equivalence class of the element relative. The set of equivalence classes is called also factor quantity.

If you have now given a function, a corresponding function on the factor set results in accordance with the rule

Means of the induced map.

Well- definedness induced in pictures

In order to show the well- definedness of, is to prove that the value is independent of the chosen representatives of the equivalence class. In other words, must apply:

Examples of induced pictures

And is in the first example. As an equivalence relation, we select the " equivalence modulo 3", that is, it applies

The equivalence relation is the ordinary equality, so in case. As a function we choose

The induced map is then

It is now, though. In this case, the induced map is not well defined.

The second example is. The equivalence relation is explained by

And is the usual equality again. The real cosine now induces the mapping

This map is well defined, how to show the following:

Be with the property. According to the definition of a by now exists, and, therefore, follows, where we have used the fact that the cosine has a period of.

Induced link

Definition of the induced link

Let be a nonempty set with an equivalence relation and an inner join. The assignment

Induced on the associated factor structure of a link.

Well- definedness of induced links

In this case, the result should not depend on the choice of representatives in a class. The link is so well-defined, if it is ensured that different representatives of the same class yield the same result. That is, it must apply to all with the property:

Examples of induced links

• The link given by, is not well defined: it is = and =, but

• Consider the symmetric group and in the subgroup. The induced on the quotient set link is not well defined. It is, and of course, but

• The addition and the multiplication in the remainder class ring are well-defined. The residue classes is just the addition of the addition and in the normal subgroup induced link.

• Is a normal subgroup of the group, then the well-defined on induced shortcut, and is called the factor group of after. The property of being normal subgroup is even equivalent to that induced link is well defined on the quotient set. Because let and arbitrary. For the well- definedness of the induced group link on the left cosets must apply:

Completeness and consistency

In a broader sense, well- definedness is extended to other areas. Then referred to a meaningful and consistent definition. Synonym for " not well defined " in this sense are also " not defined" or " not complete " used.

Definition of a function

If you have, for example, the formula, the zero must not be included in the domain, as provides for the formula. To divide by zero, however, is not explained in the real numbers, i.e., there is no real number " ." ( In a broader sense, one could indeed set. That does the example but nothing since there is no real number! Addition to and identify with each other would have, since diverges for against. )

Likewise, it is not explained in the real numbers, to take the square root of negative numbers. In other words, the " function " is not well defined, the function would be does.

Range of a function

If we write the formula as "function ", so the value of the value is assigned though. Which is in this case not admissible, since not a natural number and is thus not within the value range.

Links with groups

Inner joins an algebraic structure (eg, a group) functions are also (usually with two arguments ). For them, therefore, the same conditions apply: The combination of elements of the structure must be a uniquely determined element of result. Here the term seclusion is often incorrectly used, but which refers to the definition of sub-structures.

Well- definedness of quantities

A lot is well defined when the Definiens for any object clearly specifies that it is either element of the set or not is member of the set. In particular, as certain forms are excluded imprädikativer definitions.