THEORY CONTENT OF BASICS

Natural numbers : The numbers {1, 2, 3, 4, . . . . . } are called natural or counting numbers. The set of natural numbers are denoted by N i.e., N = {1, 2, 3, 4, . . . . .}.

(ii) Natural numbers are also called positive integers (denoted by Z⁺ or I⁺).

(iii) Whole numbers:W = {0, 1, 2, 3, 4, . . . . . }

Whole numbers are also called non negative integers.

(iv) Integers : The numbers 0, ±1, ±2, ………. are called integers. The set of integers is denote by Z or I i.e., Z = {. . . . . –3, –2, –1, 0, 1, 2, 3, . . . . .}

Zero is neither positive nor negative but it is non–positive as well as non-negative integer.

(v) Rational numbers :

Any number of the form p/q where p, qÎZ such that p and q are coprime, and q¹ 0 (because division by zero is not defined) is called a rational number.

All integers are rational numbers where q = 1

When q¹ 1 and p, q have no common factor except 1, (where p and q are both positive) the rational numbers are called fractions.

Rational numbers when represented in decimal form are either ‘terminating’ or ‘non-terminating’ but repeating.

e.g., 5/4 = 1.25 (terminating in zeros)

5/3 = 1.6666 . . . . . (non terminating but repeating)

(vi) Irrational numbers:

Numbers, which cannot be represented in form (where and ) are called irrational numbers.

In decimal representation, they are neither terminating nor repeating hence, all surds fall in this category

e.g., , e, etc.

Note : is only an approximate value of p in terms of rational numbers, taken for convenience.

Actually p = 3.14159 . . . . . (which is a non terminating and non repeating decimal number)

(vi) Real numbers:

All rational and irrational numbers taken together form the set of real numbers, represented by R. This is the largest set in the real world of numbers.

Also note that, integers which give an integer on division by 2 are called even integers otherwise they are called odd integers.

Zero is considered an even number (because there is always one even integer between two consecutive odd numbers).
The set of natural numbers can be divided in two ways.

(i) Odd and even natural numbers.

(ii) Prime numbers (which are not divisible by any number except 1 and itself) and composite numbers (which have some other factor apart from 1 and itself).

1 is neither prime nor composite.
2 is the only even number which is prime all other primes are odd.

(vii) Complex numbers:

The numbers of the form where and a, b are real numbers are known to be complex numbers.

Question: Prove that there is not a single natural number N with sum of its digits equal to 15, that is the square of an integer.

Solution: Let there be a natural number N with sum of its digits equal to 15 such that , where m is an integer, consider two cases:

(i) m is divisible by 3

(ii) m is not divisible by 3

In case (i) : m = 3p, where p is an integer.

Then is divisible by 9. But N with sum of digits equal to 15, is not divisible by 9.

In case (ii): since m is not divisible by 3, also is not divisible by 3.

But N with sum of digits equal to 15 is divisible by 3.

Thus in either case there is a contradiction which proves the desired result.

Note: Any integer may be regarded as a number which is either divisible by 3 or not divisible by 3.

1.1 Set Theory

1.1.1 Basic Concept

Set: (i) A set is a well–defined collection of objects or elements. Each element in a set is unique. Usually but not necessarily, a set is denoted by a capital letter i.e., A, B, . . . . . U, V etc. and the elements are enclosed between brackets { . }. The elements are denoted by small letters a, b, . . . . . x, y etc. For example:

A = Set of all small English alphabets

= {a, b, c, . . . . . x, y, z}

B = Set of all positive integers less than or equal to 10

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

R = Set of real numbers

= {x : – ¥< x <¥, }

The elements of a set can be discrete (e.g. set of all English alphabets) or continuous (e.g. set of real numbers). The set may contain finite or infinite number of elements. A set may contain no elements and such a set is called Void set or Null set or empty set and is denoted by f (phi) or {}.The number of elements of a set A is denoted by n(A) and hence n(f) = 0 as it contains no element.

For example: The collection of girls students in a boys college is the empty set.

(ii) A set is called a finite set if the process of counting of its different elements comes to an end and vice versa.

For example: The set of all divisors of a given natural number is a finite set.

1.1.2 Union of sets

Union of two or more sets is the set of all elements that belong to any of these sets. The symbol used for union of sets is .

i.e., = Union of set A and set B = {x : x or x }

e.g. If A = {1, 2, 3, 4} and B = {2, 4, 5, 6} and C = {1, 2, 6, 8},

then AÈBÈC = {1, 2, 3, 4, 5, 6, 8}

1.1.3 Intersection of sets

It is the set of all the elements, which are common to all the sets under consideration . The symbol used for intersection of sets is ‘Ç’ i.e. AÇB = {x : x and x} e.g. If A = {1, 2, 3, 4}, B = {2, 4, 5, 6} and C = {1, 2, 6, 8}, then
AÇBÇC = {2}.

Remember that, n(A ÈB) = n(A) + n(B) – n(AÇB)

{where means number of elements in A union B}

1.1.4 Difference of two sets

The difference of set A to B denoted as A – B, is the set of those elements that are in the set A but not in the set B i.e. A – B = {x : x and x}

Similarly B – A = {x : and }. In general, A – B¹B – A

e.g. If A = {a, b, c, d} and B = {b, c, e, f}, then A – B = {a, d} and B – A = {e. f}

Question: If A = {a, b, c} and B = {b, c, d}, then evaluate A ÈB, A ÇB, A – B and B – A.

Solution: AÈB = {x : xÎA or x ÎB} = {a, b, c, d}

AÇB = {x : xÎA and xÎB} = {b, c}

A – B = {x : xÎA and xÏ B} = {a}

B – A = {x : xÎB and xÏ A} = {d}

Divisibility in Integers:

Let a and b be two positive integers. Then b is divisible by a iff for some positive integer k. For example: 10 = 5.2 so 10 is divisible by 5 and 2 both but 10 is not divisible by 3, 7 etc.