Well-defined

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In mathematics, an expression is well defined or unambiguous if the definition provides an interpretation or a unique value. Otherwise, the expression is said to be undefined or ambiguous . A function is well defined if it gives the same result when the input representation is changed without changing the input value. For example if f takes the real number as input, and if f (0,5) is not equal f (1/2) then f is not well-defined (and therefore: not a function). The term well defined is also used to indicate whether the logical statement is not ambiguous.

Undefined functions are not the same as undefined functions. For example, if f ( x ) = 1/ x , then f (0) is undefined, but this has nothing to do with the question of whether f ( x ) = 1/ x is well defined. This; 0 is not in the function domain.

Video Well-defined

Contoh

Biarkan ${\ displaystyle A_ {0}, A_ {1}}  ÂÂ$ menjadi set, biarkan ${\ displaystyle A = A_ {0} \ cup A_ {1}}  ÂÂ$ dan "define" ${\ displaystyle f: A \ rightarrow \ {0,1 \}}  ÂÂ$ sebagai ${\ displaystyle f (a) = 0}  ÂÂ$ jika ${\ displaystyle a \ in A_ {0}}  ÂÂ$ dan ${\ displaystyle f (a) = 1}  ÂÂ$ jika ${\ displaystyle a \ in A_ {1}}  ÂÂ$ .

Kemudian ${\ displaystyle f}  ÂÂ$ didefinisikan dengan baik jika ${\ displaystyle A_ {0} \ topi A_ {1} = \ emptyset}  ÂÂ$ . Ini adalah g. kasus ketika ${\ displaystyle A_ {0}: = \ {2,4 \}, A_ {1}: = \ {3,5 \}}  ÂÂ$ (lalu f ( a ) terjadi menjadi ${\ displaystyle \ operatorname {mod} (a, 2)}  ÂÂ$ ).

To avoid apostrophes around "define" in the previous simple example, the "definition" of ${\ displaystyle f}$ can be split into two simple logical steps:

While the definition in step 1. is formulated with any freedom of definition and is certainly effective (without having to classify it as, well defined), the statement in step 2. must be proven: If and only if ${\ displaystyle A_ {0} \ cap A_ {1} = \ emptyset}$ , we get the function ${\ displaystyle f}$ , and ${\ displaystyle f}$ of "definitions" is well defined (as a function).

Di sisi lain: jika ${\ displaystyle A_ {0} \ topi A_ {1} \ neq \ emptyset}  ÂÂ$ maka untuk ${\ displaystyle a \ dalam A_ {0} \ topi A_ {1}}  ÂÂ$ ada keduanya, ${\ displaystyle (a, 0) \ in f}  ÂÂ$ dan ${\ displaystyle (a, 1) \ in f}  ÂÂ$ , dan relasi biner ${\ displaystyle f}  ÂÂ$ bukan fungsional sebagaimana didefinisikan dalam relasi Biner # Jenis hubungan biner khusus dan dengan demikian tidak terdefinisi dengan baik (sebagai fungsi). Bahasa sehari-hari, "fungsi" ${\ displaystyle f}  ÂÂ$ disebut ambigu pada titik ${\ displaystyle a}  ÂÂ$ (meskipun ada per definitionem tidak pernah ada "fungsi ambigu"), dan "definisi" asli tidak ada gunanya.

Apart from this delicate logical problem, it is very common to anticipate using the term definition (without apostrophes) for this type "definition", first because it is a kind of two-step approach, second because it is relevant mathematics. reasoning (step 2.) is the same in both cases, and finally because in the mathematical text the statement is  «to 100% » true.

Maps Well-defined

Independence representative

The question of a well defined function classically arises when the equation defines a function not (only) referring to the argument itself, but (also) to the argument element. This is sometimes unavoidable when the argument is coset and the equation refers to coset representation.

Works with one argument

Sebagai contoh, perhatikan fungsi berikut

${\ displaystyle {\ begin {matrix} f: & amp; \ mathbb {Z}/8 \ mathbb {Z} & amp; \ to & amp; \ mathbb {Z}/4 \ mathbb {Z} \\ & amp; {\ overline {n}} _ {8} & amp; \ mapsto & amp; {\ overline {n}} _ {4}, \ end {matrix}}}  ÂÂ$

di mana ${\ displaystyle n \ in \ mathbb {Z}, m \ in \ {4,8 \}}  ÂÂ$ dan ${\ displaystyle \ mathbb {Z}/m \ mathbb {Z}}  ÂÂ$ adalah modul integer m dan ${\ displaystyle {\ overline {n}} _ {m}}  ÂÂ$ menunjukkan kelas kesesuaian n mod m .

N.B.: ${\ displaystyle {\ overline {n}} _ {4}}  ÂÂ$ adalah referensi untuk elemen ${\ displaystyle n \ in {\ overline {n}} _ {8}}  ÂÂ$ , dan ${\ displaystyle {\ overline {n}} _ {8}}  ÂÂ$ adalah argumen f .

Fungsi f didefinisikan dengan baik, karena

${\ displaystyle n \ equiv n '\ operatorname {mod} 8 \; \ Leftrightarrow \; 8 \ mid (n-n') \; \ Leftrightarrow \; 2 \ cdot 4 \ mid (n-n ') \; \ Rightarrow \; 4 \ mid (n-n') \; \ Leftrightarrow \; n \ equiv n '\ operatorname {mod} 4.}  ÂÂ$

Operasi

Specifically, well-defined terms are used in respect of the (binary) operation of the coset. In this case one can see the operation as a function of two variables and the well-defined property is the same as the function. For example, the addition to integer modules of some n can be defined naturally in terms of the addition of integers.

${\ displaystyle [a] \ oplus [b] = [a b]}$

Fakta bahwa ini didefinisikan dengan baik berikut dari fakta bahwa kita dapat menulis perwakilan apa pun dari ${\ displaystyle [a]}  ÂÂ$ sebagai ${\ displaystyle a kn}  ÂÂ$ , di mana k adalah bilangan bulat. Karena itu,

${\ displaystyle [a kn] \ oplus [b] = [(a kn) b] = [(a b) kn] = [a b ] = [a] \ oplus [b];}  ÂÂ$

dan juga untuk perwakilan apa pun dari ${\ displaystyle [b]}  ÂÂ$ .

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Notasi yang terdefinisi dengan baik

Untuk bilangan real, produk ${\ displaystyle a \ times b \ times c}  ÂÂ$ tidak ambigu karena ${\ displaystyle (a \ times b) \ kali c = a \ kali (b \ times c)}  ÂÂ$ . (Oleh karena itu, notasinya dikatakan terdefinisi dengan baik .) Karena sifat operasi ini (di sini ${\ displaystyle \ times}  ÂÂ$ ), yang dikenal sebagai associativity, hasilnya tidak bergantung pada urutan perkalian, sehingga spesifikasi urutannya dapat dihilangkan.

Pengoperasian pengurangan, ${\ displaystyle -}  ÂÂ$ , tidak asosiatif. Namun, ada konvensi (atau definisi) dalam ${\ displaystyle -}  ÂÂ$ operasi dipahami sebagai penambahan sebaliknya, sehingga ${\ displaystyle a-b-c}  ÂÂ$ sama dengan ${\ displaystyle a (- b) (- c)}  ÂÂ$ , dan disebut "terdefinisi dengan baik".

Divisi juga non-asosiatif. Namun, dalam kasus ${\ displaystyle a/b/c}  ÂÂ$ konvensi ${\ displaystyle/b: = * b ^ {- 1}}  ÂÂ$ tidak begitu mapan, sehingga ungkapan ini dianggap tidak jelas .

Unlike functions, the notation ambiguity can be solved easily or more easily by using additional definitions, ie e. rules are preferred, and/or associativity of the operator. In the programming language C e.Ã, g. operator - for subtraction is left-to-right-associative which means abc is defined as (ab) -c and the = operator for assignment is right-to-left-associative which means that a = b = c is defined as a = ( b = c) . In the APL programming language there is only one rule: from right to left - but the first bracket.

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Other uses of the term

A solution to partial differential equations is said to be well defined if it is determined by continuous boundary conditions as the boundary conditions change.

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See also

• Equivalence relationships Ã,§Definition is well under the equivalent relation
• Definitionism
• Existence
• Uniqueness
• Quantify uniqueness
• Not specified

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References

Note

Source

• Contemporary Abstract Algebra , Joseph A. Gallian, Issue 6, Houghlin Mifflin, 2006, ISBNÃ, 0-618-51471-6.
• Algebra: Chapter 0 , Paolo Aluffi, ISBN 978-0821847817. Page 16.
• Abstract Algebra , Dummit and Foote, 3rd ed., ISBN 978-0471433347. Page 1.

Source of the article : Wikipedia