Algorithmic theory of integers: Różnice pomiędzy wersjami
Z Lem
| (Nie pokazano 2 pośrednich wersji utworzonych przez tego samego użytkownika) | |||
| Linia 1: | Linia 1: | ||
| − | 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: | + | Integer numbers form a set, usually denoted '''integer''' (or <math>Z</math>). The set contains a non-empty subset <math>N</math> (of non-negative integer numbers ''aka'' 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>, | ||
| Linia 9: | Linia 9: | ||
<math>a+0=a\qquad </math> and <math>\qquad a\cdot 1= a</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, | * (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, | ||
| − | * ( | + | * (Additive inverse) For every integer number <math>a</math>, exists integer number <math>-a</math>, such that the following equality holds |
<math> a+-a =0 </math> | <math> a+-a =0 </math> | ||
| − | * ( | + | * (Trichotomy) For every integer number <math>a</math> exactly one of three relations holds: either a) number <math>a</math> is non-negative, or b) number <math>a</math> is zero <math>a=0</math> or c) number <math>-a</math> jis non-negative, |
| − | * ( | + | * (Non-negative integers) The set of non-negative integers satisfies the axioms of natural numbers, it is the same set <math>N</math>. See [[Algorithmic_theory_of_natural _numbers]].<br /> |
| − | * ( | + | Note, axiom S of natural numbers is an algorithmic formula. |
| − | + | * (Ordering relation) a < b if and only if the number <math> b+-a </math> belongs to the set <math>N</math>. | |
Aktualna wersja na dzień 12:02, 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 non-negative integer numbers aka 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,
- (Additive inverse) For every integer number [math]a[/math], exists integer number [math]-a[/math], such that the following equality holds
[math] a+-a =0 [/math]
- (Trichotomy) For every integer number [math]a[/math] exactly one of three relations holds: either a) number [math]a[/math] is non-negative, or b) number [math]a[/math] is zero [math]a=0[/math] or c) number [math]-a[/math] jis non-negative,
- (Non-negative integers) The set of non-negative integers satisfies the axioms of natural numbers, it is the same set [math]N[/math]. See Algorithmic_theory_of_natural _numbers.
Note, axiom S of natural numbers is an algorithmic formula.
- (Ordering relation) a < b if and only if the number [math] b+-a [/math] belongs to the set [math]N[/math].
