Target amount

Figure 1: A function from to . ${\displaystyle A}$ ${\displaystyle B}$

In mathematics , a function that maps the elements of a set to elements of a set is called the target set or value set of the function. ${\displaystyle f\colon A\to B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle B}$

The word set of values or range of values is often used for this ; however, these words often designate the image set of . So there is a risk of confusion. In Germany there is clarity in school lessons; only the term set of values (or range of values ) is used in the sense of the image set. The target quantity is only the reserve for possible values ​​of ; it is not absolutely necessary that these are actually all accepted by. ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle f}$

The set of values ​​that appear as the function value of is the image set. If the image set of is equal to the target set of , then it is called surjective (right total). ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle f}$

The target set is a distinguishing component of a function. Functions with the same definition range and the same functional rule, but different target set, are not the same.

example

The function assigns each point of the Euclidean plane its distance from the zero point. The following applies: ${\displaystyle f\colon \mathbb {R} ^{2}\to \mathbb {R} }$

${\displaystyle f(a,b)={\sqrt {a^{2}+b^{2}}}}$

The target set of the function is the set of real numbers . Since the distance can never be negative, not all possible values ​​are assumed. The image set consists exactly of the nonnegative real numbers (often referred to as). ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} _{0}^{+}}$

The function with has the same domain, the same function rule and the same image set as . However, since the target quantities are different, it still applies . ${\displaystyle g\colon \mathbb {R} ^{2}\to \mathbb {R} _{0}^{+}}$ ${\displaystyle g(a,b):={\sqrt {a^{2}+b^{2}}}}$ ${\displaystyle f}$ ${\displaystyle f\neq g}$

Relationship between the quantities

Using the simple example function from Figure 1, the different quantities that occur should be explained again:

• The definition set ( ) contains the elements 1, 2, 3, 4 . ${\displaystyle A}$
• The target set ( ) contains the elements a, b, c, d . ${\displaystyle B}$
• The image set consists of the elements b, c, d . Only these three are actually taken as function values.
• Domain is another word for set of definitions.
• Value stock is another word for target quantity.
• The meaning of the set of values and the range of values is not clearly defined and can designate the target set or the image set. In Germany there is clarity in school lessons, only the designator value set (value range) is used in the sense of the image set.

Individual evidence

1. ↑ Harro Heuser, Textbook of Analysis. Part 1. 8th edition, BG Teubner, Stuttgart 1990. ISBN 3-519-12231-6 . P. 104.
2. ^ G. Wittstock lecture notes for Analysis 1 (PDF; 365 kB) winter semester 2000-2001. Description 1.3.3, p. 19
3. ↑ ^{a b} Reinhard Dobbener: Analysis . Oldenbourg Wissenschaftsverlag 2007, ISBN 3486579991 . S 12, definition 1.12.
4. ↑ Andreas Gathmann: Lecture Fundamentals of Mathematics , Chapter 1 Something Logic and Set Theory ( Memento from March 4, 2016 in the Internet Archive ) page 13, definition 1.18.
5. ↑ Michael Ruzicka: Analysis I . Lecture from the winter semester 2004/2005. S 21 (PDF; 74 kB).
6. ↑ Harro Heuser, Textbook of Analysis. Part 1. 8th edition, BG Teubner, Stuttgart 1990, ISBN 3-519-12231-6 , p. 106.
7. ↑ Hlawka, Binder Schmitt: Basic Concepts of Mathematics. Prugg Verlag Vienna 1979, p. 27.