1. Introduction
• The theory of sets was developed
by German mathematician George
Cantor.
• A set is a collection of objects.
• Objects in the collection are called
elements of the set.
• They are named by capital English
alphabet.
2. Representation Of
Sets• Roster form and Set Builder form
• Roster Form- when the elements are
written inside the set It is defined as
a set by actually listing its elements,
for example, the elements in the set A
of letters of the English alphabet can
be listed as A={a,b,c,……….,z} separated
by comas.
3. • Set Builder Form- when we write a
set in a straight form using
underlying relations that binds
them.
• Example- {x | x < 6 and x is a
counting number} in the set of all
counting numbers less than 6. Note
this is the same set as {1,2,3,4,5}.
5. Empty Sets
• A set that contains no members is
called the empty set or null set .
• For example, the set of the months
of a year that have fewer than 15
days has no member .Therefore ,it
is the empty set. The empty set is
written as { } or .
6. Finite Sets
• A set is finite if it consists of a
definite number of different
elements ,i.e., if in counting the
different members of the set, the
counting process can come to an end.
• For example, if W be the set of
people living in a town, then W is
finite.
7. Infinite Sets
• An infinite set is a set that is not
a finite set. Infinite sets may
be countable or uncountable. Some
examples are:
• The set of all integers, {..., -1, 0, 1, 2,
...}, is a count ably infinite set;
8. Equal Sets
• Equal sets are sets which have the
same members. Or Two sets a and b
are said to be equal if they have the
same no of elements.
• For example, if
P ={1,2,3},Q={2,1,3},R={3,2,1}
then P=Q=R.
9. Subsets
• Sets which are the part of another set
are called subsets of the original set.
• For example, if
A={1,2,3,4} and B ={1,2}
then B is a subset of A
it is represented by .
10. Power Sets• If ‘A’ is any set then one set of all are subset of set ‘A’ that
it is called a power set.
• Example- If S is the set {x, y, z}, then the subsets of S are:
• {} (also denoted , the empty set)
• {x}
• {y}
• {z}
• {x, y}
• {x, z}
• {y, z}
• {x, y, z}
• and hence the power set of S is {{}, {x}, {y}, {z}, {x, y}, {x, z},
{y, z}, {x, y, z}}.
11. Universal Sets
• A universal set is a set which contains all
objects, including itself. Or
• In a group of sets if all the sets are the
subset of a particular bigger set then that
bigger set then that bigger set is called the
universal set.
• Example- A={12345678}
B={1357}
C={2468}
D={2367}
Here A is universal set and is
denoted by
12. Operation Of Sets
• Union of sets
• Intersection of sets
• Compliments of sets
13. Union
• The union of two sets would be
wrote as A U B, which is the set of
elements that are members of A or
B, or both too.
• Using set-builder notation,
A U B = {x : x is a member of A or
X is a member of B}
14. Intersection
• Intersection are written as A ∩ B,
is the set of elements that are in A
and B.
• Using set-builder notation, it would
look like:
A ∩ B = {x : x is a member of A and
x is a member of B}.
15. Complements
• If A is any set which is the subset of a
given universal set then its complement is
the set which contains all the elements
that are in
but not in A.
• Notation A’
={1,2,3,4,5}
A={1,2,3}
A’={2,4}
16. Some Other Sets
• Disjoint – If A ∩ B = 0, then A and B
are disjoint.
• Difference: B – A; all the elements in B
but not in A
• Equivalent sets – two sets are
equivalent if n(A) = n(B).
17. Venn Diagrams
• Venn diagrams are
named after a English
logician, John Venn.
• It is a method of
visualizing sets using
various shapes.
• These diagrams
consist of rectangles
and circles.