Basic set Concepts

1. Set Concepts
By
Nutan Prakash
Assistant Professor
Department of Biotechnology
Shree M. & N. Virani Science College

2. INDEX
1. Definition of set
2. Properties of sets
3. Set theory
4. Venn Diagram
5. Set Representation
6. Types of Sets
7. Operation on Sets

3. Set
Georg Cantor a German Mathematician born in
Russia is creator of set theory
A set is a well defined collection of objects.
Individual objects in set are called as elements of set.
e. g.
Collection of even numbers between 10 and 20.
Collection of flower or bouquet.

4. Properties of Sets
1. Sets are denoted by capital letters.
Set notation : A, B, C ,D
2. Elements of set are denoted by small letters.
Element notation : a,d,f,g
For ex: Set A= {x,y,v,b,n,h}
3. If x is element of A we can write as x∈A i.e. x
belongs to set A.

5. 4. If x is not an element of A we can write as x∉A i.e
x does not belong to A
e.g If A is a set of days in a week then
Monday ∈ A and January ∉ A
5. Each element is written once.
6. Order of element is not important. i.e set A can be
written as { 1,2,3,4,5,} or as {5,2,3,4,1}

6. Set Representation
There are two main ways of representing sets.
1. Roaster method or Tabular method or
Listing Method.
2. Set builder method or Rule method

7. A. Roster or Listing method
All elements of the sets are listed, each element
separated by comma(,) and enclosed within brackets { }.
Ex:
 Set C= {1,6,8,4}
 Set T = {Monday, Tuesday, Wednesday, Thursday,
Friday, Saturday}
• Set K = {a, e, i, o, u}

8. (a) The set of all natural numbers which divide 42 is
{1, 2, 3, 6, 7, 14, 21, 42}.
• In roster form, the order in which the elements are
listed is immaterial. Thus, the above set can also be
represented as {1, 3, 7, 21, 2, 6, 14, 42}.
(b) The set of all vowels in the English alphabet is {a,
e, i, o, u}.

9. (c) The set of odd natural numbers is represented by {1, 3,
5, . . .}.
• The dots tell us that the list of odd numbers continue
indefinitely.
• It may be noted that while writing the set in roster form an
element is not generally repeated, i.e., all the elements are
taken as distinct.
• For ex: The set of letters forming the word ‘SCHOOL’ is { S, C,
H, O, L} or {H, O, L, C, S}.

10. 2. Rule method or set builder
method
• All elements of set posses a common property. e.g.
set of natural numbers is represented by
• K= {x|x is a natural no}
Here | stands for ‘such that’
‘:’ can be used in place of ‘|’
e.g. Set T= {y | y is a season of the year}
Set H = {x | x is blood type}

11.  In set-builder form, all the elements of a set possess a
single common property which is not possessed by any
element outside the set.
 For ex: in the set {a, e, i, o, u}, all the elements possess
a common property, namely, each of them is a vowel in
the English alphabet, and no other letter possess this
property.
 Denoting this set by V, we write
V = {x : x is a vowel in English alphabet}

12. • “the set of all x such that x is a vowel of the English
alphabet”. In this description the braces stand for “the
set of all”, the colon stands for “such that”. For ex:
the set A = {x : x is a natural number and 3 < x < 10}
• is read as “the set of all x such that x is a natural number
and x lies between 3 and 10”. Hence, the numbers 4, 5, 6,
7, 8 and 9 are the elements of the set A.

13. • If we denote the sets described in (a), (b) and (c)
above in roster form by A, B, C, respectively, then A,
B, C can also be represented in set-builder form as
follows:
• A= {x : x is a natural number which divides 42}
• B= {y : y is a vowel in the English alphabet}
• C= {z : z is an odd natural number}

14. • Ex: Write the set {x : x is a positive integer
and x2
< 40} in the roster form.
• Solution: The required numbers are 1, 2, 3, 4,
5, 6. So, the given set in the roster form is {1,
2, 3, 4, 5, 6}.

15. Ex: Write the set A = {1, 4, 9, 16, 25, . . . } in
set-builder form.
• Solution: We may write the set A as
A = {x : x is the square of a natural number}
Alternatively, we can write A = {x : x = n2
,
where n ∈ N}

16. Ex: Match each of the set on the left described in the roster form with
the same set on the right described in the set-builder form :
(i) {P, R, I, N, C, A, L} (a) { x : x is a positive integer and is a divisor of
18}
(ii) { 0 } (b) { x : x is an integer and x2
– 9 = 0}
(iii) {1, 2, 3, 6, 9, 18} (c) {x : x is an integer and x + 1= 1}
(iv) {3, –3} (d) {x : x is a letter of the word PRINCIPAL}
Solution: Since in (d), there are 9 letters in the word PRINCIPAL and
two letters P and I are repeated, so (i) matches (d). Similarly, (ii)
matches (c) as x + 1 = 1 implies x = 0. Also, 1, 2 ,3, 6, 9, 18 are all
divisors of 18 and so (iii) matches (a). Finally, x2
– 9 = 0 implies x = 3,
–3 and so (iv) matches (b).

17. Cardianility of set
• Number of element in a set is called as
Cardianility of set.
Number of elements in set n (A)
e.g Set A= {he, she, it, the, you}
Here no. of elements are n |A|=5
• Singleton set containing only one elements e.g
Set A={3}

18. Types of set
1.Equal set
2.Empty set
3.Finite set and Infinite set
4.Equivalent set
5.Subset Universal set

19. Equal sets
• Two sets k and R are called equal if they have
equal numbers and of similar types of elements.
• For e.g. If K={1,3,4,5,6}
R={1,3,4,5,6}
• then both Set K and R are equal.
• We can write as Set K = Set R

20. Empty sets
• A set which does not contain any elements is called
as Empty set or Null or Void set. Denoted by ∅ or { }
• e.g. Set A= {set of months containing 32 days}
Here n (A)= 0; hence A is an empty set.
• e.g. set H={no. of cars with three wheels}
Here n (H)= 0; hence it is an empty set.

21. Finite set
• Set which contains definite no of element.
e.g. Set A= {♣,♦,♥,♠}
• Counting of elements is fixed.
Set B = { x | x is no of pages in a particular book}
Set T ={ y | y is no. of seats in a bus}
Infinite set
• A set which contains indefinite numbers of elements.
Set A= { x | x is a of whole numbers}
Set B = {y | y is point on a line}

22. Subset
• A set A is said to be a subset of a set B if every element of A
is also an element of B.
• Ex: If
A= {3,5,6,8} and B = {1,4,9}
then B is a subset of A it is represented as B A⊂
• Every set is subset of itself i.e A ⊂ A
• Empty set is a subset of every set. i.e ∅ ⊂ A
.3
.5
.6.
.8
.1
.4
.9
A
B

23. Consider the sets:
X = set of all students in your school,
Y = set of all students in your class.
• We note that every element of Y is also an element of X; we
say that Y is a subset of X. The fact that Y is subset of X is
expressed in symbols as Y X. The symbol stands for ‘⊂ ⊂ is a
subset of’ or ‘is contained in’.

24. • In other words, A B if whenever⊂ a A, then a B. It is often∈ ∈
convenient to use the symbol “ ” which means⇒ implies. Using
this symbol, we can write the definition of subset as follows:
A B if⊂ a A a B∈ ⇒ ∈
• We read the above statement as “A is a subset of B if a is an
element of A implies that a is also an element of B”. If A is
not a subset of B, we write A B.⊄

25. Universal set
• A set U that includes all of the elements under
consideration in a particular discussion. Depends on the
context. It is designated by the symbol U.
Ex: The set of Latin letters,
set of natural numbers, the set of points on a line.
• e.g. Set T = {The deck of ordinary playing cards}. Here each
card is an element of universal set.
Set A= {All the face cards}
Set B= {numbered cards}

26. Venn Diagrams
Most of the relationships between sets can be represented by means
of diagrams which are known as Venn diagrams. Venn diagrams are
named after the English logician, John Venn (1834-1883).
These diagrams consist of rectangles and closed curves usually circles.
The universal set is represented usually by a rectangle and its
subsets by circles. In Venn diagrams, the elements of the sets are
written in their respective circles.

27. Operation on Sets
• Intersection of sets
• Union of sets
• Difference of two sets
• Complement of a set

28. Intersection of sets
Let A and B be two sets. Then the set of all
common elements of A and B is called the
Intersection of A and B and is denoted by A∩B
 Let A={1,2,3,7,11,13}}
B={1,7,13,4,10,17}}
 Then a set C= {1,7,13}} contains the elements
common to both A and B
Hence A B is represented by shaded part in venn∩
diagram.
Thus A B= { x | x∩ ∈A and x ∈B}

29. Union of sets
• Let A and B be two given sets then
the set of all elements which are in
the set A or in the set B is called the
union of two sets and is denoted by A
U B and is read as ‘A union B’
Union of Set A and Set B= {1, 2, 3, 4,0, 2, 4, 5, 6}

30. Difference of two sets
1.The difference of set A- B is set of all
elements of “A” which does not belong to
“B”.
2.In set builder form difference of set is:
A-B = {x: x∈A x∉B}
B-A = {x: x ∈B x∉A}
e.g Set A = {1,4,7,8,9}
Set B = {3,2,1,7,5}
Then A-B = {4,8,9}

31. Disjoint sets
• Sets that have no common members are called
disjoint sets.
• Ex: Given that
U= {1,2,3,4,5,6,7,8,9,10}
• Set A = {1,2,3,4,5}
• Set C = {8,10}
• No common elements hence set A and C are
disjoint set.

32. Complements
o If A is a subset of the universal set U, then the
complement of A is the set
o Note:
Φ=∩ c
AA UAA c
=∪
{ }c
A x U x A= ∈ ∉
The shaded region represents
the complement of the set A

33. • Ex: Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A =
{1, 3, 5, 7, 9}. Find A .′
• Soln: We note that 2, 4, 6, 8, 10 are the only
elements of U which do not belong to A.
• Hence A = { 2, 4, 6, 8,10 }.′

34. • Ex: Let U be universal set of all the students
of Class XI of a coeducational school and A be
the set of all girls in Class XI. Find A .′
• Solution: Since A is the set of all girls, A is′
clearly the set of all boys in the class.

35. Applications
1.A set having no element is empty set. ( yes/no)
2.A set having only one element is singleton set. (yes/no)
3.A set containing fixed no of elements.{ finite/ infinite set)
4. Two set having no common element. (disjoint set
/complement set)