Natural Numbers

•We sat two sets A and B are equinumerous if there exists a map
f : A −→ B which is bijective : one-to-one and onto

• The relation equinumerous is an equivalence relation.

It decomposes any class of sets into disjoint equivalence classes. Two
sets belong to the same equivalence class if they are equinumerous.

• The characteristic property of each equivalence class is defined to be

the number of elements of that class

•The set of numbers of elements may be defined as the set of natural

numbers N

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 Natural Numbers
Axioms of Peano

There exists an injective map : F : N → N which maps any

element n ∈ N to n+ such that n6=m ⇒ n+ 6= m+ .

This map is called the successor map and n+ is called the successor
of n.

There exists one element called 1 such that 1 6= n+ ∀ n ∈N

Suppose that A ⊆N and (i) 1 ∈ A (ii) n+ ∈ A whenever n ∈ A .

Then A =N .

The only subset of N which contains 1 and the successor of all its
elements is itself N

This third axiom is called the Principle of Mathematical Induction.

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 Natural Numbers : Addition
Axioms for Addition

If m, n ∈ N , then we define addition inductively as follows :

(i) m + 1 = m+ and (ii) m + n+ = (m + n)+ .

If A ={n, for which n + m is defined as above}, then if n ∈ A then

n+ ∈ A . Also 1 ∈ A since 1 + n is defined. So A =N .

The axioms of Addition are :

1. Commutativity : n + m = m + n

2. Associativity : n + (m + p) = (n + m) + p

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 Natural Numbers : Multiplication
Axioms for Multiplication

If m, n ∈ N , then we define multiplication inductively as follows :

(i) m.1 = m and (ii) m.n+ = m.n + m.

The axioms of Multiplication are :

1. Commutativity : m.n = n.m

2. Associativity : m.(n.p) = (m.n).p

3. Distributive : m.(n + p) = m.n + m.p

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 Natural Numbers

If m, n ∈ N , then we define m < n, if ∃ k ∈ N : m + k = n

A non-empty set A is called a finite set if there exists an n ∈ N and a

bijective map F such that :

F : A → {1, 2, . . . n}

A set A is defined to be an infinite set if and only if it has a proper subset

to which it is equinumerous
Dedekind’s criterion

Example : N and N -{1} are equinumerous, since there exists a

bijective map : F : {n → n + 1}

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 Integers

A few more axioms define the negative numbers and zero and the
concept of subtraction

1. Existence of Additive identity ∀ n ∈ N , there exists an integer 0 such

that n + 0 = n.

2. Existence of Additive inverses (Negatives) ∀ n ∈ N there exists

another n such that n+n=0.

Obviously 0 or n6∈ N . The set Z =N ∪ {0} ∪ {n} is defined to be the set

of integers

n is defined as the integer −n

Subtraction is defined as n − m = n + (−m)

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 Integers

These axioms define the the concept of division

1. Existence of Multiplicative identity ∀ n ∈ N , n.1 = n

2. Existence of Multiplicative inverses (Reciprocals) ∀ n ∈ N there

exists another ñ such that n.ñ = 1

ñ is defined as the reciprocal n−1 or 1/n

Division is defined as n/m = n.m−1

If we leave out the element 0 ∈ Z , then all of the above is also

defined on Z

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 Theorems

1. If n, m, p ∈ Z and n + p = n + q → p = q. This implies that 0 is

unique

2. Given n, m ∃ a unique p such that n + p = m. This unique p is

called m − n.

3. n.(m − p) = n.m − n.p

4. 0.n = n.0 = 0

5. If n.m = 0, then n = 0 or m = 0 or both.

6. Given n, m 6= 0 ∈ Z , ∃ a unique p such that n.p = q. In particular,

n.1 = n so 1 is unique

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 Rational Numbers

Let Z∈ = Z ⊗ Z -{0}. That is, Z∈ ={(m, n), m, n ∈ Z n 6= 0}. Define an

equivalence relation ∼ by (m, n) ∼ (m0 , n0 ) if mn0 = m0 n. This
equivalence relation decomposes Z∈ into disjoint equivalent classes
containing (m, n). We shall label these equivalence classes by m n
.

We may define addition and multiplication among members of different

equivalence classes as :

1) If m, n, m0 , n0 ∈ Z and n, n0 6= 0 then :

m m0 mn0 + m0 n
+ 0 =
n n nn0

2) If m, n, m0 , n0 ∈ Z and n, n0 6= 0 then :

m m0 m.m0
· =
n n0 n.n0

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 Rational Numbers

3) If m, n, n0 ∈ Z and n, n0 6= 0 then :
m 0 m
+ 0 =
n n n
So 0
n0
behaves like the additive identity 0.

4) If m, n, n0 ∈ Z and n, n0 6= 0 then :

m n0 m
· 0 =
n n n

So n0
n0
behaves like the multiplicative identity 1.

5) If m, n ∈ Z and m, n 6= 0 then :
m n
· = 1
n m
So n
m
is the multiplicative inverse of m
n
.

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 Rational Numbers

We call the set of all equivalence classes m

n
the set of rational numbers :
m
Q = {α = ; m, n ∈ Z, n 6= 0}
n

Nine properties are satisfied by members of Q :

1) α + β = β + α

2) α + (β + γ) = (α + β) + γ

3) α + 0 = α

4) ∀α ∈Q ∃ a unique β ∈Q such that α + β = 0.

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 Rational Numbers

5) α.β = β.α

6) α.(β.γ) = (α.β).γ

7) α.1 = α

8) α.(β + γ) = α.β + α.γ

9) ∀α ∈Q -{0}, ∃ a unique β ∈Q such that α.β = 1

Any set with two binary operations (+,.) with all nine properties is called
a Field. Set Q is a field. But not N or Z .

If α, β ∈Q , then we define α < β if ∃ a γ ∈ Q + such that β + γ = α.

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 Rational Numbers : Inequalities
1. If α, β ∈ Q and γ ∈ Q + : α < β ⇒ αγ < βγ

2. If α, β ∈ Q + and α < β ⇒ β −1 < α−1

3. If α, β ∈ Q and γ ∈ Q − : α < β ⇒ βγ < αγ

4. If α, β ∈ Q ⇒ −β < −α.

Theorem 1 Archimedean property of Q

If α ∈ Q + and β ∈ Q , ∃ n ∈ N such that β < nα.

Proof : If β < α then n = 1. If α < β set α = p/q and β = p0 /q 0 , p, q, p0 , q 0 ∈ N

-{0}.

αq = p. Let p0 /q 0 = n1 + r0 /q 0 , where n1 ∈ N ∪ {0} and 0 ≤ r 0 < q or 0 ≤ r 0 /q 0 < 1

which implies that we cal always find a m ∈N such that mp > n1 + r0 /q 0 .

Now set qm = n so that nα = mqα = mp > p0 /q 0 = β QED

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 Rational Numbers
Is α such that α2 = 2 a rational number ?

Suppose α ∈ Q , then write α = m/n ⇒ m2 = 2n2 . Also suppose that

m, n have no common factors. Then 2 is a factor of m2 and hence of m,
so m = 2p. So, 2p2 = n2 and hence 2 is a factor of n. Consequently, 2 is
a common factor of both m and n. This is a contradiction. Thus γ 6∈ Q .

How do we extend Q to include such irrational numbers ?

Upper Bound : a is an upper bound for the set B if ∀ b ∈ B , b < a

Least Upper Bound : a is the least upper bound of a set B if

(i) a is an upper bound of B
(ii) if a0 < a, then a0 cannot be an upper bound of B
We call a = sup B

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 Rational Numbers
redCompleteness Axiom : Every non-empty set which is bounded
above has a supremum

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