Sets: Definition and Examples
In Mathematical language all living and non-living things in universe are
known as objects. The collection of well defined distinct objects is known as
a set.
Well defined means in a given set, it must be possible to decide whether or not
the object belongs to the set and by distinct means object should not be
repeated. The object in the set is called its member or element. A set is
represented by { }.
Generally, sets are denoted by capital letters A B C , , , . . . and its elements are
denoted by small letters a b c , , , . . . . .
Let A is a non-empty set. If x is an element of A, then we write ‘x $\in$ A’ and
read as ‘x is an element of A’ or ‘x belongs to A’. If x is not an element of A,
then we write ‘x $\in$ A ’ and read as x is not an element of A’ or ‘x does not belong
to A’.
e.g., A = Set of all vowels in English alphabets.
In this set a e i o u , , , and are members.
Sample Problem 1
Which of the following is a correct set?
(a) The collection of all the months of a year beginning with the letter J.
(b) The collection of ten most talented writers of India.
(c) A team of eleven best cricket batsmen of the world.
(d) A collection of most dangerous animals of the world.
Interpret:
(a) We are sure that members of this collection are January, June and July.
So, this collection is well defined and hence, it is a set.
(b) A writer may be most talented for one person and may not be for other. Therefore, it
cannot be said accountely that which writers will be there in the relation.
So, this collection is not well defined. Hence, it is not a set
(c) A batsman may be best for one person and may not be so
for other. Therefore, it cannot be said accountely that
which batsman will be there in our relation.
So, this collection is not well defined. Hence, it is not a set.
(d) The term most dangerous is not a clear term. An animal
may be most dangerous to one person and may not be for
the other.
So, it is not well defined, hence it is not a set.
Sample Problem 2
Verify whether the following are sets:
(1) The collection of all intelligent persons in Visakhapatnam.
(2) The collection of all prime ministers of India.
(3) The collection of all negative integers.
(4) The collection of all tall persons in India.
Interpret:
Note that the collections given in (1) and (4) are not
sets because, if we select a person in Visakhapatnam,
we cannot say with certainty whether he/she belongs to
the collection or not, as there is no stand and scale for the
evaluation of intelligence or for being tall. However, the
collections given in (2) and (3) are sets.
Sample Problem 3
The set of intelligent students in a class is
(a) a null set
(b) a singleton set
(c) a finite set
(d) not a well defined collection
Interpret:
The set of intelligent students in a class is not a well defined
collection, because intelligency is not defined for a student
in a class
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