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** SET THEORY**

**Definition of a set :**

A set is any
collection of distinct and distinguishable objects around us. By the form
‘distinct’, we mean that no object is repeated and some lack the term
‘distinguishable’ we mean that whether that object is in our collection or not.
The objects belonging to a set are called as elements or members of that set.

Ex: A is a set of
stationary used by an student i.e.,

A= {Pen, Pencil, Eraser, Paper}

B= {Hyderabad, Chennai, Mumbai}

Here B is also a
set of Capitals of 3 States.

A set is
represented by using all its elements between braces { } and by separating them
from each other by commas (if there are more than one element).

The sets are
denoted by capital letters of English alphabet while the elements are divided
in general, but small letters.

If x is an element
of a set A, we write xA (read as ‘X belogins to A’). If x is not
an element of A. we write x not A
(read as x does not belong to A).

**(i)**Let A={6, 8, 6, 3, 5}. The elements of this collection are disinguishable but not distinct. Hence A is not a set.

**(ii)**Let B={a, e, I, o, u} i.e., B is set of vowels in English. Hence elements of B is distinguishable as well as distinct. Hence B is a set.

**Two forms of Representation of a set:**

**(i)**‘Set-builders form’ representation of set, and

**(ii)**‘Tabular form’ or ‘Roaster form’ representation of set.

**Set-bulider form :**

In
‘Set-builder form’ of representation of set, we write between the braces () a
variable x which stands for each of the elements of the set, then we state thae
properties possessed by x. we denote this property by p(x) and separte x and
p(x) by a symbol: or (read as ‘such that’).

A= { x: p (x) }

A={x: x is capital of
States}

Or
A={ x: x is a natural number and 2<x<11}

**Tabular Form :**

In
‘Tabular Form’ or ‘Roaster Form’ the elements of a set are listed one by one
within braces { } and one separaed by each other by commas.

B={ Hyderabad, Chennai, Mumbai }

Or B={
3, 4,5, 6, 7, 8, 9, 10 }

**Order of a set:**

The
number of distinct elements of a set is
called order of a set. If ‘A’ is set, then its order is denoted by n(A).

**Equal sets:**

Two
sets having the same elements are called equal sets.

Example
: If A={1,2,3,4} and B={1,2,3,4}, the A
and B are equal sets

**Equivalent sets:**

Two
sets having the same number of elements are called equivalent sets.

Example
: If A={ a,b,c,d} and B ={1,2,3,4}, the A and B are equivalent sets

**Types of sets:**

**1.**

**Null Set (or Empty Set or Void Set):**

A set having no elements is called an
empty set or void set. It is denoted by Ø or { }.

**Ex:**

**A = {x: x is an even number not divisible by 2} or**

B={x: x is a even prime
number greater than 2}

**2.**

**Singleton Set:**

A set having a single element is called a singleton
set.

Ex: A={x: x is even prime numbers} or

B={a} or

C={x: x

^{2}=x , x is a positive number }**3.**

**Pair Set:**

**A set having two elements is called a pair set.**

Ex: {1, 2}, {0, 3}, {4, 9}. Etc.

**4.**

**Finite Set**

**:**

A set having a finite number of elements i.e., a set,
where counting of elements is possible is called a finite set.

**Ex:**A= {1, 2, 4, 6} is a finite set because it has four elements.

A null set Ø is also a finite set because it has zero
number of elements.

C={x: x is a factor of 2000}

**5.**

**Infinite Set:**

A set having infinite number of elements i.e., a set
where counting of elements is impossible, is called an infinite set.

**Ex:**A= {x: x is a set of all natural numbers}

**B ={ x:**x is a multiple of 12}

**SUBSETS, SUPERSETS, AND PROPER SETS:**

Set A is said to be a subset of a set B if
each element of set A is also an element of set B.

If A is a subset of set B, we represent it
as A Í B.

So, if A Í B
ó {xA => xB }

Ex:
Let A={4, 5, 6} and B={ 4, 5, 6, 7, 8, 6}, then we write A Í B.

Let’s
see another example.

A={1, 2, 3}, B={2, 3, 1} => A subset B and
B subset A

Here
A Í B
can also be expressed equivalently by writing B Ê A, read as B is subset of
another set B, if and only if set B contains all the elements of set A.

**Proper Subsets:**

**Set A is said to be a proper subset of a set B if**

**(a)**Every element of set A is an element of set B, and

**(b)**Set B has at least one element which is not an element of set A.

This is expressed by writing A Ì B or B Ì A and read as a A is a
proper subset of B or B is a proper subset of A, if A is not a proper subset of
B then we write it as A Ë B.

**(i)**Let A ={4, 5, 6} and B ={4, 5, 6, 7, 8}

**(ii)**Let A ={ 1, 2, 3} and B={3, 2, 9}

**Trivial and Non-Trivial Subsets:**

For
any ‘A’, the empty set and ‘A’ are always subsets which are called trivial
subsets.

If any other subsets of ‘A’, which are
not real are called ‘non-trivival’ subsets of ‘A’.

If A is Finite set such that n(A)=n
then

**(i)**The no. of subsets of ‘A’ are 2

^{n}.

**(ii)**The no. of trivial subsets of ‘A’ are 2.

**(iii)**The no. of non-trivial subsets of ‘A’ are 2

^{n}-2.

**(iv)**The no. of improper subsets of ‘A’ are 1.

**(v)**The no. of proper subsets of ‘A’ are 2

^{n}-1