2. A set is a collection of well defined objects. The objects are
well distinguished also.
eg., a set of all natural numbers from1 to 50.
The numbers are (i) well defined
(ii) Well distinguished
The objects that constitutes set are called its members or
elements.
3. Description of Set
A set is described in the following two ways:
(i)Roster method: Under this method a set is described by
listing elements, seperated by commas, within braces{ }.
Eg., The set of vowels of English Alphabet may be
described as {a,e,i,o,u}.
The order in which the elements are written in a set does
not make any difference.
Therefore, {a,e,i,o,u} and {e,a,i,o,u} denote the same set.
4. (ii) Set-Builder method:Under this method , a set is
described by a characterizing property P(x) of its elements
x.
In such a case the set is described by
{x:P(x) holds} or, {x│P(x) hols}, which is read as “the
set of all x such that P(x) holds”. The symbol ‘│, or ‘:’ is
read as’such that’.
Eg., The set A={1,2,3,4,5,6,7} can be written as A={x ∈N
:x≤7}
5. Types of Sets
Empty Set: A set is said to be empty or null or void
set if it has no element and it is denoted by ∅.
A set consisting of atleast one element is called a non
empty or non-void set.
Singleton set: A set consisting of a single element is
called a singleton set .
Eg., the set{6} is a singleton set.
6. Finite Set: A set is called a finite set it its is either void set
or its elements can be listed, counted,labelled by natural
number ‘n’ (say).
Cardinal Number of a Finite Set: The number ‘n’ in the
above definition is called the cardinal number or order of a
finite set A and is denoted by n(A).
Infinite Set: A set whose elements cannot be listed by the
natural numbers 1,2,3,….n for any natural number n is
called an infinite set.
7. Equivalent Set: Two finite sets A and B are
equivalent if their cardinal numbers are same i.e.,
n(A) = n(B).
Equal Sets: Two sets A and B are said to be equal
if every element of A is a member of B and every
element of B is a member of A.
If sets are equal we write A=B and A≠B , when A
and B are not equal.
8. Subsets: Let A and B be two sets. If every element of A is
an element of B, then A is called a subset of B. If A is a
subset of B, we write A ⊆ B, which is read as “ A is a
subset of B” or “A is contained in B”.
Thus, A ⊆ B if a ∈A ⇒ a ∈ B.
If A is a subset of B, we say that B contains A or B is a
Super set of A and we write B ⊃A.
If A is not a subset of B, we write A ⊄ B.
Every set is a subset of itself and the empty set is
subset of every set. These two subsets are called
improper subsets.
9. Universal Set:
In set theory, there is a set that contains all sets under
consideration i.e., it is a super set of each of the given sets.
Such a set is called the Universal set and is denoted by U.
In other words, a set that contains all sets in a given
context is called the universal set.
If A={1,2,3,4}, B={2,3,4,5} and C= {1,3,5,7}, the
U={1,2,3,4,5,6,7} can be taken as the universal set.
10. Power set: Let A be a set. Then the collection of
family of all subsets of A is called the power set of A
and is denoted by P(A).
Let A={1,2,3}. Then the subsets of A are:
∅,{1},{2},{3},{1,2},{1,3},{2,3} and {1,2,3}.
Hence P (A)=
{∅, ,{1},{2},{3},{1,2},{1,3},{2,3} {1,2,3}}.
The number of proper subsets in a finite set is obtained
by n(⊂)=2ⁿ -1 where, n represents the number of
elements in a finite set and n(⊂)represents the number
of proper subsets of the said set.
