Definition set

The definition of this quantity function X → Y is {1, 2, 3} , in this case, the entire base set X .

In mathematics , the definition set or definition domain means the set with exactly those elements under which (depending on the context) the function is defined or the statement can be fulfilled. In school mathematics, the definition set is often abbreviated with , sometimes it is also written with a double line . ${\ displaystyle D}$${\ displaystyle \ mathbb {D}}$

Domain of definition of a function

A function is a special relation that assigns exactly one element of the target set to each element of the definition set . The definition set is denoted by. If the function has a different name than e.g. B. or , then the definition area is referred to as or . ${\ displaystyle f \ colon A \ to B}$ ${\ displaystyle A}$ ${\ displaystyle B}$${\ displaystyle D_ {f}}$${\ displaystyle f}$${\ displaystyle g}$${\ displaystyle h}$${\ displaystyle D_ {g}}$${\ displaystyle D_ {h}}$

The amount

${\ displaystyle \ {f (a) \ mid a \ in A \} = \ {b \ mid \ exists a \ in A \ colon f (a) = b \} \ subseteq B}$

of all function values of is called the image or set of values of and is a subset of the target set. ${\ displaystyle f (a)}$${\ displaystyle f}$ ${\ displaystyle W_ {f}}$${\ displaystyle f}$

The basic set and the target set of a function are essential parts of its definition. However, the basic set and the target set of a function are often not specified if the function is meant on the maximum possible definition set (which is then usually a subset of the real numbers or complex numbers ). ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle \ mathbb {C}}$

However, two functions with the same functional dependency but different basic sets or different target sets are different functions and can have different properties.

Examples

The mapping with the basic set and the target set is given . Then: is a function with and . ${\ displaystyle f \ colon x \ mapsto x ^ {2}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle f}$${\ displaystyle D_ {f} = \ mathbb {R}}$${\ displaystyle W_ {f} = \ mathbb {R} _ {0} ^ {+}}$

1. As a function (i.e. with definition set and target set ) is bijective , i.e. both surjective and injective .${\ displaystyle \ mathbb {R} _ {0} ^ {+} \ to \ mathbb {R} _ {0} ^ {+}}$${\ displaystyle \ mathbb {R} _ {0} ^ {+}}$${\ displaystyle \ mathbb {R} _ {0} ^ {+}}$${\ displaystyle f}$
2. As a function (i.e. with definition set and target set ) is injective, but not surjective.${\ displaystyle \ mathbb {R} _ {0} ^ {+} \ to \ mathbb {R}}$${\ displaystyle \ mathbb {R} _ {0} ^ {+}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle f}$
3. As a function (i.e. with definition set and target set ) is surjective, but not injective.${\ displaystyle \ mathbb {R} \ to \ mathbb {R} _ {0} ^ {+}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R} _ {0} ^ {+}}$${\ displaystyle f}$
4. As a function (i.e. with definition set and target set ) is neither surjective nor injective.${\ displaystyle \ mathbb {R} \ to \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle f}$

Restriction and continuation of a function

Be a function and , . The function is called restriction of if applies to all . in this situation means extension or continuation of . ${\ displaystyle f \ colon A \ to B}$${\ displaystyle U \ subseteq A}$${\ displaystyle V \ subseteq B}$${\ displaystyle r \ colon U \ to V}$${\ displaystyle f}$${\ displaystyle r (x) = f (x)}$${\ displaystyle x \ in U}$${\ displaystyle f}$${\ displaystyle r}$

The restriction is often written as. This notation is not completely exact because the amount is not given; in the most interesting cases, however, the choice is usually made . ${\ displaystyle r}$${\ displaystyle r = f \ left | _ {U} \ right.}$${\ displaystyle V}$${\ displaystyle V = B}$

For a function and two given sets , there is at most one restriction of ; this exists if and only if the image set is a subset of . ${\ displaystyle f \ colon A \ to B}$${\ displaystyle U \ subseteq A}$${\ displaystyle V \ subseteq B}$${\ displaystyle r \ colon U \ to V}$${\ displaystyle f}$${\ displaystyle f}$${\ displaystyle V}$

In contrast to the restriction of a function, the continuation is not clear.

example

The function is given

${\ displaystyle f \ colon \ left \ {{\ begin {matrix} \ mathbb {R} \ setminus \ {0 \} \ to \ mathbb {R} \\ x \ mapsto x \ sin {\ frac {1} { x}} \ end {matrix}} \ right.}$

Possible extensions to the domain of definition , i.e. as functions , are, for example, both ${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R} \ to \ mathbb {R}}$

${\ displaystyle f_ {0} \ colon x \ mapsto \ left \ {{\ begin {matrix} 0 & {\ mbox {if}} x = 0 \\ x \ sin {\ frac {1} {x}} & { \ mbox {falls}} x \ neq 0 \ end {matrix}} \ right.}$

as well as

${\ displaystyle f_ {1} \ colon x \ mapsto \ left \ {{\ begin {matrix} 1 & {\ mbox {if}} x = 0 \\ x \ sin {\ frac {1} {x}} & { \ mbox {falls}} x \ neq 0 \ end {matrix}} \ right.}$

${\ displaystyle f_ {0}}$is a "nicer" continuation insofar as is steady , but not. However, this does not change the fact that both functions are correct continuations, since a clear continuation is not preserved in the function definition itself. Uniqueness only results from additional requirements, such as continuity in this example, or for example in the requirement for a holomorphic continuation to the complex numbers of a function that is initially only defined on a subset of the real numbers. ${\ displaystyle f_ {0}}$ ${\ displaystyle f_ {1}}$

Domain of definition of a relation

Under the domain of the relation with ${\ displaystyle R = (A, B, G)}$

${\ displaystyle G \ subseteq A \ times B}$

one understands the projection of on , i.e. that subset of elements of the source that appear as the first components in elements : ${\ displaystyle R}$${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle (a, b) \ in G}$

${\ displaystyle \ operatorname {Dom} (R): = \ {a \ in A \ mid \ exists b \ in B: (a, b) \ in G \}.}$

example

The relation is given with ${\ displaystyle R = (\ mathbb {R}, \ mathbb {R}, G)}$

${\ displaystyle G = \ {(x, y) \ mid x = y ^ {2} \}}$.

Since for real the square is always non-negative (greater than or equal to zero), and vice versa for each non-negative real least a real number with exists, for this relation of the definition area, the amount of non-negative real numbers . ${\ displaystyle y}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle x = y ^ {2}}$${\ displaystyle \ operatorname {Dom} (R) = \ mathbb {R} _ {0} ^ {+}}$

Domain of definition of a term

The domain of a term with variables and related basic quantities is the set of all n-tuples , for , for the term goes into meaningful values. ${\ displaystyle n}$${\ displaystyle a_ {i}}$ ${\ displaystyle A_ {i}}$ ${\ displaystyle (\ alpha _ {1}, \ dots, \ alpha _ {n})}$${\ displaystyle \ alpha _ {i} \ in A_ {i}}$${\ displaystyle i = 1, \ dots, n}$

Examples

The domain of the term in a variable with the basic set is that the fraction is only meaningfully defined for a non-zero value of the denominator. ${\ displaystyle {\ frac {1} {x-1}}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ {x \ in \ mathbb {R} \ mid x \ neq 1 \}}$

The domain of the term in two variables with the basic set is because in the real case the root is only meaningfully defined for non-negative values. ${\ displaystyle {\ sqrt {x-1}} + {\ sqrt {y-2}}}$${\ displaystyle \ mathbb {R} \ times \ mathbb {R}}$${\ displaystyle \ {(x, y) \ in \ mathbb {R} \ times \ mathbb {R} \ mid x \ geq 1 {\ mbox {and}} y \ geq 2 \}}$

Domain of definition of equations and inequalities

Are and terms are called ${\ displaystyle T_ {1}}$${\ displaystyle T_ {2}}$

${\ displaystyle T_ {1} = T_ {2}}$

an equation ,

${\ displaystyle T_ {1} \ leq T_ {2}}$

and

${\ displaystyle T_ {1}> T_ {2}}$

and similar expressions are called inequalities . When solving an equation or inequality, one looks for those values ​​from the basic range for which the equation or inequality turns into a true statement. The domain of definition is the subset of the basic domain for which all terms appearing in the equation or inequality are meaningfully defined, i.e. the average set of the defined set of and . ${\ displaystyle T_ {1}}$${\ displaystyle T_ {2}}$

Particularly in the case of more complicated equations, it can happen that when solving the initial equation it is converted to an equation that also contains solutions that are not contained in the definition range of the initial equation. In such a case, after solving the equation, it must be checked whether the solution values ​​obtained are actually contained in the definition range and, if necessary, some values ​​are eliminated.

example

They are the real solutions of the equation

${\ displaystyle {\ sqrt {x-1}} + {\ sqrt {x + 1}} = {\ sqrt {2x + {\ frac {3} {2}}}}}$

searched. Since only non-negative values ​​are allowed under the root, this is the domain of the equation . ${\ displaystyle \ {x \ in \ mathbb {R} | x \ geq 1 \}}$

Squaring the equation yields

${\ displaystyle 2x + 2 {\ sqrt {x ^ {2} -1}} = 2x + {\ frac {3} {2}}}$

or.

${\ displaystyle {\ sqrt {x ^ {2} -1}} = {\ frac {3} {4}}}$.

Squaring is not an equivalent conversion , it is true , but not , so the converted equation can contain more solutions than the output equation. Squaring again gives ${\ displaystyle a = b \ implies a ^ {2} = b ^ {2}}$${\ displaystyle a ^ {2} = b ^ {2} \ implies a = b}$

${\ displaystyle x ^ {2} -1 = {\ frac {9} {16}}}$

or.

${\ displaystyle x ^ {2} = {\ frac {25} {16}}}$.

This equation has two solutions and . The value is not in the domain of the equation and is therefore not a solution; when inserted into the output equation, the value yields a true statement and is therefore the only solution to the equation. ${\ displaystyle x = - {\ frac {5} {4}}}$${\ displaystyle x = {\ frac {5} {4}}}$${\ displaystyle x = - {\ frac {5} {4}}}$${\ displaystyle x = {\ frac {5} {4}}}$

Individual evidence

1. Edmund Hlawka , Christa Binder , Peter Schmitt: Basic concepts of mathematics. Prugg Verlag, Vienna 1979. ISBN 3-85385-038-3 . P. 38 f, definition 3.13.
2. ^ A b Walter Gellert, Herbert Kästner , Siegfried Neuber (eds.): Lexicon of Mathematics. VEB Bibliographisches Institut Leipzig, 1979. p. 167, function VII.
3. Edmund Hlawka, Christa Binder, Peter Schmitt: Basic concepts of mathematics. Prugg Verlag, Vienna 1979. ISBN 3-85385-038-3 . P. 39, Theorem 3.13 and Theorem 3.14.
4. Edmund Hlawka, Christa Binder, Peter Schmitt: Basic concepts of mathematics. Prugg Verlag, Vienna 1979. ISBN 3-85385-038-3 . P. 18.
5. ^ Walter Gellert, Herbert Kästner, Siegfried Neuber (eds.): Lexikon der Mathematik , VEB Bibliographisches Institut Leipzig, 1979. P. 199, equation.