The document defines and explains various concepts related to sets. It discusses how sets were developed by Georg Cantor and defines elements and representation of sets using roster and set builder forms. It also defines and provides examples of empty, finite, infinite, equal, subset, power and universal sets. Finally, it explains operations of sets such as union, intersection and complements and provides examples of each.

2. 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.

3. 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.

4. • 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. Types Of Sets
• Empty Sets
• Finite Sets
• Infinite Sets
• Equal Sets
• Subsets
• Power Sets
• Universal Sets

6. 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 .

7. 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.

8. 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;

9. 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.

10. 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 .

11. 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}}.

12. 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

13. Operation Of Sets
• Union of sets
• Intersection of sets
• Compliments of sets

14. 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}

15. 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}.

16. 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}

17. 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).

18. 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.