Question

You have studied a number of mathematical structures. Vector space, metric space, topo- logical space, group, ring and field are some examples. Give general definitions and specific examples. Comment on some of the details of some of these structures. Explain how various kinds of functions are involved in these structures.

Answer 1

VECTOR SPACE:----- A vector space (over ) consists of a set along with two operations subject to these conditions.

1. For any .
2. For any .
3. For any .
4. There is a zero vector such that for all .
5. Each has an additive inverse such that .
6. If is a scalar, that is, a member of and then the scalar multiple is in .
7. If and then .
8. If and , then .
9. If and , then
10. For any , .

example:-

The set of real-valued functions of the real variable is a vector space under the operations

and inherited from the space in the prior example. (We can think of as "the same" as in that corresponds to the vector with and )

METRIC SPACE;----

A metric space is a set X together with a function d (called a metric or "distance function") which assigns a real number d(x, y) to every pair x, y X satisfying the properties (or axioms):

1. d(x, y) 0 and d(x, y) = 0 x = y,
2. d(x, y) = d(y, x),
3. d(x, y) + d(y, z) d(x, z).

Examples

1. The prototype: the line R with its usual distance d(x, y) = |x - y|.
2. The plane R² with the "usual distance" (measured using Pythagoras's theorem):
   d((x₁ , y₁), (x₂ , y₂)) = [(x₁ - x₂)² + (y₁ - y₂)²].
   This is sometimes called the 2-metric d₂ .

TOPO-LOGICAL SPACE:-----  

Let X be a set. A set of subsets of X is called a topology (and the elements of are called open sets) if the following properties are satisfied.

1. (the empty set), X ,
2. if {A_i | i I} then A_i,
3. if A, B then A B .

Examples

1. The prototype
   Let X be any metric space and take to be the set of open sets as defined earlier. The properties verified earlier show that is a topology.
2. Some "extremal" examples
   Take any set X and let = {, X}. Then is a topology called the trivial topology or indiscrete topology.
   Let X be any set and let be the set of all subsets of X. The is a topology called the discrete topology. It is the topology associated with the discrete metric.

GROUP:----

A group is a set G together with an operation that takes two elements of G and combines them to produce a third element of G.The operation must also satisfy certain properties.

More formally, the group operation is a function G*G->G , which is denoted by(x,y)->x*y , satisfying the following properties (also known as the group axioms).

Group Axioms:1) Associativity: For any x,y,z in G , we have(x*y)*z=x*(y*z) .

2) Identity: There exists an e in G such that e*x=x*e=x for any x in G . We say that e is an identity element of G .

3) Inverse: For any x in G , there exists y in G such that x*y=e=y*x . We say that y is an inverse of x.

Note that the definition of the operation as a function implies

4) Closure: For any x,y in G, x*y is also in G.

EXAMPLES:-----

1) Z,the set of integers, with the group operation of addition.

2) R*,the set of non-zero real numbers, with the group operation of multiplication.

3) Zn,the set of integers{0,1,...n-1} , with group operation of addition modulo n.

RING:-----

A ring is a set R together with two operations (+) and (.) satisfying the following properties (ring axioms):

(1) R is an abelian group under addition. That is, R is closed under addition, there is an additive identity (called 0), every element a in R has an additive inverse -a in R, and addition is associative and commutative.

(2) R is closed under multiplication, and multiplication is associative:

for all a,b in R => a.b in R

for all a,b,c in R => a.(b.c)=(a.b),c

(3) Multiplication distributes over addition:

for all a,b,c in R => a.(b+c)=a.b+a.c and (b+c).a=b.a+c.a

A ring is usually denoted by (R,+,.) and often it is written only as R when the operations are understood.

EXAMPLE:-

1.The ring Z of integers is the canonical example of a ring. It is an easy exercise to see that Z is an integral domain but not a field.

2.If Z is a ring, then so is the ring R[x] of polynomials with coefficients in R. In particular, when R=Z/pZ is the finite field with p elements,R[x] has many similarities with Z. For example, there is a Euclidean algorithm and hence unique factorization into irreducibles. See the introduction to algebraic number theory for details.

FIELD:------

A field is a set F together with two operations called addition and multiplication. An operation is a mapping that associates an element of the set to every pair of its elements. The result of the addition of a and b is called the sum of a and b and denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, and denoted ab or a⋅b. These operations are required to satisfy the following properties, referred to as field axioms. In the following definitions, a, b and c are arbitrary elements of the field F.

• Associativity of addition and multiplication: a + (b + c) = (a + b) + c and a · (b · c) = (a · b) · c.
• Commutativity of addition and multiplication: a + b = b + a and a · b = b · a.
• Additive and multiplicative identity: there exist two different elements 0 and 1 in F such that a + 0 = a and a · 1 = a.
• Additive inverses: for every a in F, there exists an element in F, denoted −a, called additive inverse of a, such that a + (−a) = 0.
• Multiplicative inverses: for every a ≠ 0 in F, there exists an element in F, denoted by a⁻¹, 1/a, or 1/a, called the multiplicative inverse of a, such that a · a⁻¹ = 1.
• Distributivity of multiplication over addition: a · (b + c) = (a · b) + (a · c) .

EXAMPLE:-----

1.Rational numbers

Rational numbers have been widely used a long time before the elaboration of the concept of field. They are numbers which can be written as fractions a/b, where a and b are integers, and b ≠ 0. The additive inverse of such a fraction is −a/b, and the multiplicative inverse (provided that a ≠ 0) is b/a, which can be seen as follows:

The abstractly required field axioms reduce to standard properties of rational numbers. For example, the law of distributivity can be proven as follows:

2.Real and complex numbers

The real numbers R, with the usual operations of addition and multiplication, also form a field. The complex numbers C consist of expressions

a + bi

where i is the imaginary unit, i.e., a (non-real) number satisfying i² = −1. Addition and multiplication of real numbers are defined in such a way that all field axioms hold for C. For example, the distributive law enforces

(a + bi)·(c + di) = ac + bci + adi + bdi², which equals ac−bd + (bc + ad)i.

The complex numbers form a field. Complex numbers can be geometrically represented as points in the plane, and addition resp. multiplication of such numbers then corresponds to adding resp. rotating and scaling points. The fields of real and complex numbers are used throughout mathematics, physics, engineering, statistics, and many other scientific disciplines.

3.A field with four elements

Addition	Multiplication
+ O I A B O O I A B I I O B A A A B O I B B A I O	· O I A B O O O O O I O I A B A O A B I B O B I A

In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. The following example is a field consisting of four elements called O, I, A, and B. The notation is chosen such that O plays the role of the additive identity element (denoted 0 in the axioms above), and I is the multiplicative identity (denoted 1 in the axioms above). The field axioms can be verified by using some more field theory, or by direct computation. For example,

A · (B + A) = A · I = A, which equals A · B + A · A = I + B = A, as required by the distributivity.

This field is called a finite field with four elements, and is denoted F₄ or GF(4).The subset consisting of O and I (highlighted in red in the tables at the right) is also a field, known as the binary field F₂ or GF(2). In the context of computer science and Boolean algebra, O and Iare often denoted respectively by false and true, the addition is then denoted XOR (exclusive or), and the multiplication is denoted AND. In other words, the structure of the binary field is the basic structure that allows computing with bits.

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