2.
are properties or axioms
set to develop the system of
natural numbers.
by Giuseppe Peano.
(1858 – 1932, Italian)
System of Natural numbers
“ natural number
(denoted by N),”
“successor,” and “1”
3.
Postulate 1. (P5.1):
1 is a natural number, i.e. 1 Є N.
Postulate 2 (P5.2):
For each natural number n, there
exists, corresponding to it, another
natural number n* called the
successor of n.
4.
Postulate 2 (P5.2):
For each natural number n, there exists,
corresponding to it, another natural
number n* called the successor of n.
.
The successor of n, where n is a natural
number , is the number n* = n + 1, where
“+” is the ordinary addition.
5.
Postulate 3 (5.3):
1 is not the successor of any natural
number, i.e. for each n Є N, n* ≠ 1.
Postulate 4 (5.4):
If m, n Є N and m* = n*, then m = n.
6.
Axiom of Mathematical Induction
Postulate 5 (P5.5):
If S is any subset of N which is known
to have the following two properties:
i. The natural number 1 is in the set S.
ii. If any natural number k is in the set S, then
the successor k* of k must also be in the
set S.
Then S is equal to N.
8.
Addition on N
Definition 5.2. Addition on N
Since m* = m + 1, then we define
addition on N by n + m* = (n + m)*
whenever n + m, called the sum, is
defined.
9.
Axiom 1 (A5.1)
Closure Law for Addition
Given any pair of natural numbers, m and n,
in the stated order , there exists one and
only one natural number, denoted by m + n,
called the sum of m and n. The numbers m
and n are called terms of the sum.
For all m, n Є N, n + m Є N.
10.
Axiom 2 (A5.2)
The Commutative Law for Addition
If a and b are any natural numbers, then
a+b=b+a
11.
Axiom 3 (A5.3) Associative Law for Addition
If a, b, c are any natural number, then
(a + b) + c = a + (b + c)
13.
Multiplication on N
Definition 5.3 Product; Factor
If a and b are natural numbers, the
product of a and b shall mean the
number b +b +b +… +b where there are a
number of b’s in the sum.
In symbols,
ab = b +b +b +… +b
where the number of b terms on the right of
the equals sign is a. The numbers a and b
are called the factors of the product.
14.
Axiom 4 (M5.1)
Postulate of Closure for Multiplication
If a and b are natural numbers, given in
the stated order, there exists one and only
one natural number denoted by ab or (a)(b)
called the product of a and b.
15.
Axiom 5 (M5.2)
Commutative Law for Multiplication
If a and b are any natural numbers, then
ab = ba
16.
Axiom 6 (M5.3)
Associative Law for Multiplication
If a, b, c are any natural numbers, then
(ab)c = a(bc)
17.
Axiom 7 (D5.1)
Distributive Law of Multiplication over Addition
If a, b, c are natural numbers, then
a (b + c) = a • b + a • c
or
(b + c) a = b • a + c • a = a • b + a • c
18.
Definition 5.4 Similar Terms
Two terms are called similar terms
if they have a common factor.
20.
Subtraction on N
Definition 5.5 Difference
If a and b are natural numbers, the
difference of a and b is a natural number x
such that b + x = a, provided such number
exists.
• In symbos,
a – b = x iff b + x = a
•
21.
Definition 5.6 (Greater Than)
If a and b are natural numbers, then we say
that a is greater than b if there exists a
natural number x such that b + x = a.
In symbols,
a > b if there is x such that b + x = a
22.
Definition 5.7 (Less Than)
A natural number x is less than another
number y if and only if y is greater than x.
In symbols,
x < y iff y > x
23.
Division on N
Definition 5.8
Quotient; Multiple; Divisible; Factor
• If a and b are natural numbers, the quotient in
dividing a by b is a natural number x such that
bx = a, provided such a number exists.
• In symbols,
a ÷ b = x if bx = a
• In the statement bx = a, we say that a is a
multiple of b or a is divisible by b or b is a factor
of a.
24.
Theorem 5.1.
If p and q are natural numbers, then
(p + q) – p = q
25.
Theorem 5.2.
If p and q are any natural numbers, then
(pq) ÷ p = q
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