PRESENTATION
Overview of sets
and set
Operations
Presented by1.Priyankan
2.Rajat Singh
Introduction
•
•
The collection of well-defined distinct objects is known as a set. The word
well-defined refers to a specific property which makes it easy to identify
whether the given object belongs to the set or not. The word ‘distinct’ means
that the objects of a set must be all different.
Sets are denoted by uppercase letters (such as A,S,B…..) and its elements are
written by lowercase letters (such as a,b,c…..) enclosed by curly braces {} .
• For example:
1. The collection of children in class VII whose weight exceeds 35 kg
represents a set.
Examples
i) The set O of odd positive integers less than 10 can be expressed by :
O={1,3,5,7,9}
ii) The set A of positive integers less than 5 can be expressed by:
A={4,3,2,1,0}
iii) The set V of all vowels in the English alphabet can be written as ;
V={a,e,i,o,u}
SYMBOLS RELATED
TO SETS
Representation of Sets
• There are two ways to represent a set:
1. Roster form (tabular)
2. Set builder form
• In roster form elements are simply seperated by commas.
• For example
The set B of positive numbers less than 100 can be denoted by;
B={1,2,3,4,…,99}.
VENN DIAGRAMS
• Sets can be represented graphically using Venn diagrams. In
Venn diagrams the universal set U, which contains all the
objects under consideration, is represented by a rectangle. Inside
this rectangle, circles or other geometrical figures are used to
represent sets. Sometimes points are used to represent the
particular elements of the set. Venn diagrams are often used to
indicate the relationships between sets.
UNIVERSAL SET
• A universal set is a set which contains all the elements or objects
of other sets, including its own elements. It is usually denoted by
the symbol ‘U’.
Suppose Set A consists of all even numbers such that, A = {2, 4,
6, 8, 10, …} and set B consists of all odd numbers, such that, B =
{1, 3, 5, 7, 9, …}. The universal set U consists of all natural
numbers, such that, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10,….}.
Therefore, as we know, all the even and odd numbers are a part of
natural numbers. Therefore, Set U has all the elements of Set A
and Set B.
NULL SET (∅ 𝒐𝒓 {})
• There is a special set has no elements. This set is called the empty set, or null set,
and is denoted by {} or ∅ .Often, a set of elements with certain properties turns
out to be the null set. For instance, the set of all positive integers that are greater
than their squares is the null set. A set with one element is called a singleton set.
• A common error is to confuse the empty set ∅ with the set {∅}, which is a
singleton set. The single element of the set {∅} is the empty set itself! A useful
analogy for remembering this difference is to think of folders in a computer file
system. The empty set can be thought of as an empty folder and the set consisting
of just the empty set can be thought of as a folder with exactly one folder inside,
namely, the empty folder.
SUBSET
• The set A is said to be a subset of B if
and only if every element of A is also an
element
of B. We use the notation A⊆B to
indicate that A is a subset of the set B.
Venn diagram of A⊆B
SUPERSET
• A superset in math is a set of elements containing all of
the elements of another set. In other words, if
set A contains all of the elements of set B, then A is a
superset of B.
•
•
•
•
For example, consider the following two sets:
A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
B = {1, 3, 5, 7, 9}
Notice that all of the elements of B are in the set A.
Therefore, we say that A is a superset of the set B.
PROPER SUBSET
When A⊆B but A≠B, we say A is the proper sub set of B. It is denoted by
A⊂B. For A⊂B to be true, it must be the case that A⊆B and there must
exist an element x of B that is not an element of A. That is, A is a proper
subset of B if ∀𝑥 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ⋀∃𝑥 𝑥 ∈ 𝐵⋀𝑥 ∉ 𝐴 is true.
Example –
A={3,8,9} B={1,5,6} and C={1,2,4,3,5,6,8,9}
Here sets A and B both are proper subsets of set C because A⊂C and
B⊂C.
POWER SET
• Given set the power set S is the set of all subsets of the sets S. The
power set of S is denoted by P(S).
• In general, if A is a set with n(A)=M , then it can be shown that
n[P(A)]=2 .
Example 1- What is the power set of the set A={0,1,2} ?
Solution- We know that n(A)=3 so total no. of subset in power set
would be 23=8 subsets. Hence,
P(A)={∅ ,{0},{1},{2},{0,1},{0,2},{1,2},{0,1,2}}.
SET OPERATIONS
• The set operations are performed on two or more sets to obtain a
combination of elements, as per the operation performed on them. Set
operations are the operations that are applied on two more sets to develop a
relationship between them.
• There are four main kinds of set operations which are:
 Union of sets(∪)
 Intersection of sets(∩)
 Complement of a set(A’)
 Difference between sets(-)
UNION OF SETS
• Let A and B sets. The union of the sets A and
B, denoted by A ∪B, is the set that contains
those elements that are either in A or in B , or
in both.
• element belongs to the union of the sets A and
B if and only if x belongs to A or x belongs to
B. This tells us that
AUB = {x|x ∈A∨x ∈ B}.
The Venn diagram shown in Figure 1
represents the union of two sets A and B. The
area that represents A U B is the shaded area
within either the circle representing A or the
circle representing B.
INTERSECTION OF SETS
Let A and B be sets. The intersection of the sets A
and B, denoted by An B, is the set containing those
elements in both A and B.
An element x belongs to the intersection of the sets A
and B if and only if x belongs to A and x belongs to
B. This tells us that
A∩B={ x|x ∈A∧x ∈ B}.
The Venn diagram shown in Figure represents the
intersection of two sets A and B. The shaded area that
is within both the circles representing the sets A and B
is the area that represents the intersection of A and B.
INTERSECTION OF SETS
• DISJOINT SETS
Two sets are called disjoint if their
intersection is the empty set.
ExampleLet X= {2,4,6,8} and Y =
{1,3,5,7}. Because A ∩ B = ∅ , X
and Y are disjoint.
DIFFERENCE OF SETS
• Let A and B be sets. The difference of A and
B, denoted A-B, is the set containing those
elements that are in A but not in B. The
difference of A and B is also called the
complement of B with respect to A.
• An element x belongs to the difference of A
and B if and only if x ∈ A and x ∉ B. This
tells us that
A - B = {x | x ∈A ⋀x ∉ B}.
The Venn diagram shown in Figure represents
the difference of the sets A and B. The shaded
area inside the circle that represents A and
outside the circle that represents B is the area
that represents A - B.
COMPLIMENT OF A SET
Formal
definition
for
the
complement of a set: A = { x | x 
A}
Or U – A, where U is the
universal set
Further examples (assuming U = Z)
{1, 2, 3} = { …, -2, -1, 0, 4, 5, 6,
…}
SYMMETRIC DIFFRENCE
• The symmetric difference of two set A and B
denoted by A⊕B or A∆B is the set of that
belongs to A or to B, but not to both A and B.
A ⊕ B={x: x ∈ a or x ∈ B, but not both} or
A ⊕B=(A-B)U(B-A)
EXAMPLE- If A={1,2,3,4,5}, B={4,5,6,7,8,9}
Thus A⊕B=(AUB)-(A ∩B)
A ⊕B ={1,2,3,6,7,8,9}
SET IDENTITIES
De Morgan
Membership table
• Membership tables show all the
combinations of sets an element
can belong to
• 1 means the element belongs, 0 means
it does not
• Consider the following membership
table
A
1
1
0
0
B
1
0
1
0
AUB A∩B A-B
1
1
0
1
0
1
1
0
0
0
0
0
EXAMPLE ON SET IDENTITIES
APPLICATIONS OF SETS