# Integers

The opposite number of the natural number a is denoted bya.
The set of all natural numbers, zero and the opposite of natural numbers is called the set of integers and is denoted by Z, ie.

Z = {…, – 3, -2, -1, 0, 1, 2, 3, … }

Elements -1, -2, -3, … are negative integers, and natural numbers 1, 2, 3, … are positive integers. Zero is neither a positive nor a negative number.
Absolute value of an integer a, denoted by | a |, is the number itself if it is positive or zero, and is the opposite number if the given one is negative, ie.

#### 1.1. COLLECTION

If a, b ∈ No, then:

1. – ( a) = a; – (+ a) = -a
2. (+ a) + (+ b) = + (a + b)
3. (-a) + ( b) = – (a + b)
4. (+ a) + (– b) = + (a b), for a> b
5. (+ a) + (– b) = (b a), for a <b

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#### 1.2. MULTIPLICATION

If a, b ∈ No, then:

1. (+ a) · (+ b) = + (ab)
2. (-a) · (-b) = + (ab)
3. (-a) · (+ b) = – (ab)

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#### 1.3. SUBTRACTION AND DIVISION OF INTEGRAL NUMBERS

The difference between any two integers is an integer, ie.
(∀a, b ∈ Z) (a-b) ∈ Z

Subtraction of integers is defined by addition. ie
(∀a, b ∈ Z) a-b = a + (- b)

For given integers a and b, the quotient a: b is not always
is determined in Z, ie. is not an integer.

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#### 1.4. CONGRUMENTS

Let a, b be integers and m be a natural number. A and b are said to be congruent by modulus m, if a and b have the same residues when dividing by m, ie. a = mq₁ + r and b = mq₂ + r. Symbolic congruence is denoted by the following
way:
a ≡ b (mod m)

Example. 42 ≡ 98 (mod 4)
=>
42 = 4 · 10 + 2 and 98 = 4 ∙ 24 + 2

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#### 1.5. OPERATIONS. GROUPS

Let G be an empty set. An operation in Set G is a rule or rule * according to which each ordered pair of elements x, y ∈ G is joined by exactly one element
ZG, i.e. x * y = z
Each non-empty set G, together with some of its operations * is called a groupoid and is written (G, *)

Example 1.
(N, +); (N, ∙); (Z, -) are groupoids, and (N, -); (Z,:) are not groupoids.

For a groupoid (G, *) it is said that:

– is commutative if * is a commutative operation, ie. х * у = у * х, for any x, y ∈ G
– is associative (or is a subgroup), if * is an associative operation, ie. (x * y) * z = x * (y * z), for any x, у, zG.
-there is a neutral element, if there is an element is ∈ G such that x * e = e * x = e
-there is an inverse element if for every x ∈ G, there exists x ‘∈ G, such that x * x’ = x ‘* x = is (the element x is called inverse).

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#### 1.6. GROUP

A groupoid (G, *) is a group, if it meets the conditions:

1 “(G₁) (G, *) is an associative groupoid, ie. semi-group
; 2 “(G₂) (G, *) has a neutral element;
3″ (G₃): Each element in (G, *) has an inverse element;
4 “(G₄): If the groupoid (G, *) is commutative, then (G, *) is said to be a commutative group or an Abelian group.

– Celi broevi –

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