Page 33 - DMTH401_REAL ANALYSIS

Unit 2: Algebraic Structure and Countability

natural numbers, the sum or the product is a natural number. These operations of addition or Notes
multiplications on the sets of numbers are examples of a binary operation on a set. In general,
we can define a binary operation on a set in the following way:
Definition 2: Binary Operation
Given a non-empty set S, a binary operation on S is a rule which associates with each pair of
elements of S, a unique element of S.
We denote this rule by symbols such as ., *, +, etc.
By an Algebraic Structure, we mean a non-empty set together with one or more binary operations
defined on it. A field is an algebraic structure which we define, as follows:
Definition 3: Field Structure

A field consists of a non-empty set F together with two binary operations defined on it, denoted
by the symbols ‘+’(addition) and ‘.’ (multiplication) and satisfying the following axioms for any
elements x, y, z of the set F.

A : x + y  F (Additive Closure)
1
A : x + (y + z) = (x + y) + z (Addition is Associative)
2
A : x + y = y + x (Addition is Commutative)
3
A : There exists an element in F, denoted by ‘0’ and (Additive Identity)
4
called the zero or the zero element of F
such that x + 0 = 0 + x = x " x  F
A : For each x  F, there exists an element –x  F with (Additive Inverse)
5
the property
x + (– x) = (–x) + x = 0
The element – x is called additive inverse of x.

M : x.y  F (Multiplicative Closure)
1
M : (x.y).z = x. (y.z) (Multiplication is Associative)
2
M : x.y = y.x (Multiplication is Commutative)
3
M : There exists an element 1 different from (Multiplicative Identity)
4
0 called the unity of F, such that
1.x = x. 1 = x " x  F

M : For each x  F, x  0, there (Multiplicative Inverse)
5
–1
exists an element x  F such that
–l
x.x = .x x = 1.
–1
–1
The element x is called the multiplicative inverse of x.
D: x.(y + z) = x.y + x.z (Multiplication is distributive over Addition).
(x + y) z = x.z + y.z.

Since the unity is not equal to the zero i.e. 1  0 in a field, therefore any field must contain at least
two elements. Note that the axioms A (closure under addition) and M (closure under
1 1
multiplication) are unnecessary because the closures are implied in the definition of a binary
operation. However, we include them, for the sake of emphasis.

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