Algorithmic theory of integers: Różnice pomiędzy wersjami
Z Lem
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| − | + | Integer numbers form a set, usually denoted '''integer''' (or <math>Z</math>). The set contains a non-empty subset <math>N</math> (of natural numbers) and equipped with two binary operations , of addition and multiplication, usually denoted, <math>+</math> i <math>\cdot</math>. These operations fulfill the following axioms: | |
* (Commutativity) For every pair of integer numbers <math>a, b</math> the following equalities hold | * (Commutativity) For every pair of integer numbers <math>a, b</math> the following equalities hold | ||
<math>a+b=b+a\qquad</math> and <math>\qquad a\cdot b = b \cdot a</math>, | <math>a+b=b+a\qquad</math> and <math>\qquad a\cdot b = b \cdot a</math>, | ||
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* (Distributivity) For every triplet of integer numbers the following equality holds | * (Distributivity) For every triplet of integer numbers the following equality holds | ||
<math>(a+b)\cdot c =a\cdot c + b\cdot c</math> | <math>(a+b)\cdot c =a\cdot c + b\cdot c</math> | ||
| − | * ( | + | * (Units) There exist integer numbers 0 i 1 such that, for every <math>a</math> the following equalities hold |
| − | <math>a+0=a\qquad </math> | + | <math>a+0=a\qquad </math> and <math>\qquad a\cdot 1= a</math>, |
| − | * ( | + | * (Closure in <math>N</math>) If <math>a</math> and <math>b</math> are non-negative integer numbers then numbers <math>a+b</math> and <math>a\cdot b</math> are also non-negative integer numbers, |
* (Addytywna odwrotność) Dla każdej liczby całkowitej <math>a</math>, istnieje taka liczba całkowita <math>-a</math>, że zachodzi równość | * (Addytywna odwrotność) Dla każdej liczby całkowitej <math>a</math>, istnieje taka liczba całkowita <math>-a</math>, że zachodzi równość | ||
<math> a+-a =0 </math> | <math> a+-a =0 </math> | ||
Wersja z 11:29, 2 paź 2018
Integer numbers form a set, usually denoted integer (or [math]Z[/math]). The set contains a non-empty subset [math]N[/math] (of natural numbers) and equipped with two binary operations , of addition and multiplication, usually denoted, [math]+[/math] i [math]\cdot[/math]. These operations fulfill the following axioms:
- (Commutativity) For every pair of integer numbers [math]a, b[/math] the following equalities hold
[math]a+b=b+a\qquad[/math] and [math]\qquad a\cdot b = b \cdot a[/math],
- (Associativity) For every triplet of integer numbers [math]a,b, c[/math] the following equalities hold
[math](a+(b+c))=((a+b)+c) \qquad[/math] and [math]\qquad (a\cdot(b\cdot c))=((a\cdot b)\cdot c)[/math],
- (Distributivity) For every triplet of integer numbers the following equality holds
[math](a+b)\cdot c =a\cdot c + b\cdot c[/math]
- (Units) There exist integer numbers 0 i 1 such that, for every [math]a[/math] the following equalities hold
[math]a+0=a\qquad [/math] and [math]\qquad a\cdot 1= a[/math],
- (Closure in [math]N[/math]) If [math]a[/math] and [math]b[/math] are non-negative integer numbers then numbers [math]a+b[/math] and [math]a\cdot b[/math] are also non-negative integer numbers,
- (Addytywna odwrotność) Dla każdej liczby całkowitej [math]a[/math], istnieje taka liczba całkowita [math]-a[/math], że zachodzi równość
[math] a+-a =0 [/math]
- (Trichotomia) Dla każdej liczby całkowitej [math]a[/math] zachodzi dokładnie jedna z trzech relacji: albo a) liczba [math]a[/math] jest nieujemna, albo b) liczba [math]a[/math] jest zerem [math]a=0[/math] albo c) liczba [math]-a[/math] jest nieujemna,
- (Liczby całkowite nieujemne) Zbiór liczb całkowitych nieujemnych spełnia aksjomaty liczb naturalnych, to jest ten sam zbiór.
- (Definicja porządku) a < b wtedy i tylko wtedy gdy b+-a należy do [math]N[/math].
c.d.n.
