Friday, 7 August 2015

1. SET

Some Important Laws

In the presence of more than one operation, the order in which operations are performed is indicated by parenthesis. Thus, means the intersection of A with the union of B and C repeated use of the same operation is also permissible. Some of the important laws which hold between union, intersection, complement and Cartesian product are given below.



1. Associative Law







2. Distributive Law















3. Demorgan's Laws


2. REAL AND COMPLEX NUMBER SYSTEM

One of he most common uses of numbers is in counting and measuring distances. The natural numbers were known as counting numbers. People were well acquainted with the use of natural numbers and fractions for measuring distances. We are surprised to learn that a Greek intellectual, Pythagoras in the middle of the sixth century B. C., Discovered that there are fractions. For example square root 2, square root 3, etc. He felt that corresponding to every distance there ought to be a number. Symbols 0, 1, 2, 3, 9 used to denote numbers are called numerals.




3. IRRATIONAL NUMBERS

Irrational number cannot be written in p/q form  Irrational number is any real numbers that cannot be expressed as a ratio of integers. Irrational numbers cannot be represented as terminating or repeating decimals.for example: π , √2, √3, √5,  3.1416…….., etc


4. RATIONAL NUMBERS

Every rational number can be representing in the form of p/q and q is not equal to zero.Both are real number, rational numbers are frequently expressed as decimals.
For example, ¼= 0.25, 5/8= 0.625,
                        1/3= 0.3333….,                       6/11=0.5454…, etc.
The decimal said to be “terminating” if the remainder is zero, otherwise it is called “non-terminating”.
Non-terminating is of two types. If the expansion contains repeated blocks of digits, as in the case of 1/3 and 6/11, this is a rational number. If the expansion does not contain repeated blocks then the number is irrational...
The term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers.Note: In retinal numbers the common fraction is changed in decimal fractions and decimal is changed in common fractions.
Natural, Integers (positive and negative) prime, whole, odd, even and compound numbers etc are rational numbers.



5. COMPLEX NUMBERS

A complex number is a NUMBER that can be expressed in the form a + bi, where a and b are Real numbers and i is the Imaginary unit that satisfies the equation x2 = −1, that is, i2 = −1.[1] In this expression, a is the real part and b is the imaginary part of the complex number. The complex number a + bi can be identified with the point (a, b) in the complex plane
A Complex Number is a combination of a Real Number and an Imaginary Number
 Imaginary number:
The ordered pair (0, 1) is denoted by the letter i , read as “IOTA”. Then , i2 = i.i
= (0, 1) . (0, 1)
= (-1, 0) ∈ Complex
= -1 R
so .i=-1.
“i”  is called an imaginary number because there is no real number x satisfying the property  x2 = -1.
The number of  the form  ib   is called an imaginary number,b R.

Thursday, 6 August 2015

6. REAL NUMBERS

A number x is a real number if xQ or xI. Thus the set of R numbers, R=QI such that NWZQR.

The type of number we normally use such as 1, 15.83, -1.2, 4/5, etc. Positive or negative, large or small, whole numbers or decimal numbers are all Real Numbers. They are called “Real Numbers”  because they are not Imaginary Numbers.For example 1, 2, -4, π, 3, 5, 0.0003, 1/2 , etc.Real numbers have no starting or ending. There are infinite real numbers between 1 and 2 as well as 2 and 3 and so on.

Real Numbers