Algebraic Math System

Mathematics in the Modern World

Algebraic mathematical system

Presentors

Dean Robyn V. CervasBSN-1B

Kristen Claire R. BulawanBSN-1B

table of contents

4. Different Properties

1. Presentors

5. Examples

2. Mathematical System

3. Algebraic System

6. Solutions

Mathematical System

The mathematical systemhas been defined as a set of undefined terms, definitions, postulates or axioms, and propositions or theorems as in the Book I of the Elements of Euclid.

Geometryis a concrete mathematical system where the elements of the set being considered are concrete figures and how these objects of the set are related.

definition

Mathematical System

is a non-empty set S with a set of relations and operations on the elements of the set S and a set of axioms concerning the elements of S and how the elements are related to each other.

VS

Algebraic System

Abstract System

Algebraic

A mathematical system is classified as algebraic if it carries many properties of the set of real numbers. The axioms are not separate entities from the operations. In fact the properties of the operations are the axioms themselves.

an algebraic system can be defined as follows:

• S is a non-empty set:
• is a binary operation on S, and
• a family if relations governing the elements of S.

Existence of inverse element

Set of S and Binary operation

the following properties:

Commutative Property

Associative Property

Closure

Distributive Property

Existence of identity element

A set S is closed if using the binary operation * you combine two elements, the result is also in the set S. This means that for any elements a and bin S, a*b ϵ S.

Closure

The set S has identity element, say e, if for any element a in S, a*e=e*a = a.

Existence of Identity element

Existence of inverse element.

For the set S to be a mathematical system, each of the elements in S should have an inverse found in the same set. The idea of an inverse is that performing the binary operation on an element with its inverse will give the identity.

This means that for any element a and bin S, if a*b e then b is the inverse of a.

A binary operation * is commutative in the set S if performing this operation on any two elements of the set S will yield the same answer even if the order of the elements combined are changed. This that for any elements a and bin S, a*b = b*a.

Commutative property

Associative Property

A binary operation is associative in the set S if performing the operation on three elements by pairs will give the same answer even if the order of groupings by pairs is changed. To illustrate, let a, b, and e be elements of S. (a*b)*c = a*(b*c).

Given two binary operation say and *, then is distributive over if for every a, b, and c elements of S. a*(b.c) = (a*b)•(a*c).

Distributive property

Example and Solution

show that the set R of real numbers and the operations of addition and multiplication is a mathematical system.

Closure: Let a and b be real numbers, a + b is also real number.

Example:The numbers 5 and 6 are real numbers, then 5+6=11 is a real number, and 5•6=30 is also a real number.

Identity: The identity element for addition of R is zerosince any real number added to zero is the real number itself i.e., for any element, a of R then 0+a=a+0=a. The identity element of multiplication in R is 1 since for any a element of R, 1•a = a · 1 = a

Example:3 is a real number, 3 +0=3 and 3•1=3.

Inverse: Each real number a added to its negative is zero i.e., a +(-a)= -a +a=0. Also, each real number a ≠ 0, multiplied by its reciprocal is equal to one i.e.. a•1/a= 1/a•a=1. This implies that the additive inverse of a is -a while its multiplicative inverse is 1/a

Example: 4+(-4)=-4+4=0 4•1/4=1/4•4=1

Commutative: Let a, b be real numbers, then a+b=b+a, and for multiplication, a •b=b•a

Example:2+3= 3 + 2 = 5 and 2•3=3•2=6

Associative: Let a, b, and e be real numbers then (a+b)+c= a + (b+c). Formultiplication (a•b)•c=a•(b-c)

Example:(2+3)+4=2+(3+4)=9

Distributive: a•(b+c)=(a•b)+(a•c) and (b+c)•a=(b•a) + (c•a)

Example: 2(3+4)=2(3)+2(4)=14

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