Download Set Theory

Set Theory
Ankur Sinha, PhD
Production and Quantitative Methods
Indian Institute of Management
Ahmedabad
Preparatory Programme - Mathematics
Topics to be covered in the course
Set Theory
Progressions
Functions
Vectors and Matrices
Differentiation
System of Linear Equations
Theory of Quadratic Equations
Maxima and Minima
Integration
Permutations and Combinations
Probability
Preparatory Programme - Mathematics
Classes
Session 1
June 3 11:55 am -1:10 pm
Session 2
June 4 11:55 am -1:10 pm
Session 3
June 5 11:55 am -1:10 pm
Session 4
June 6 11:55 am -1:10 pm
Session 5 - 6
June 8 10:20 am -1:10 pm
Session 7
June 9 11:55 am -1:10 pm
Session 8
June 10 11:55 am -1:10 pm
Session 9
June 11 11:55 am -1:10 pm
Session 10 - 11
June 12 10:20 am -1:10 pm
Session 12
June 13 11:55 am -1:10 pm
Preparatory Programme - Mathematics
Assessment
•  No exams in the course
•  Exercises will be provided for self assessment
•  In case of doubts contact the instructor after the class
•  Personal meetings can be reserved on any day
between 15:00 to 17:00 hours (Office: Wing 4-G)
•  Please use email ([email protected]) only for
queries about the course practicalities and not for
discussing doubts or solutions to problems
Preparatory Programme - Mathematics
Sets
A set is a collection of objects or elements
Examples:
A = {apple, orange, banana, mango}
B = {1, 2, 3, 4, 5}
C = {table, chair, bench, desk}
Note:
1.  Preferably denote set with an upper case character
2.  Ordering of elements in a set is not important
Preparatory Programme - Mathematics
Notations
A = {a1, a2, a3, a4, a5}
B = {b1, b2, b3, b4}
Finite Sets
A contains 5 elements, a1, a2, a3, a4, a5
a1 ∈ A
a2 ∈ A
a3 ∈ A
a4 ∈ A
a5 ∈ A
b1 ∉ A
b1 ∈ B
b2 ∈ B
b3 ∈ B
b4 ∈ B
Preparatory Programme - Mathematics
Special Sets
Null Set: A set with no elements
Denoted as ∅
Universal Set: A set containing all
elements under consideration
Denoted as Ω
Preparatory Programme - Mathematics
Standard Sets
Natural Numbers with 0: N0 = {0, 1, 2, 3, 4, …}
Natural Numbers without 0: N+ = {1, 2, 3, 4, …}
Integers: Z = {…, -2, -1, 0, 1, 2, …}
Positive Integers: Z+ = {1, 2, 3, 4, …}
Infinite but
countable sets
Negative Integers: Z- = {…, -4, -3, -2, -1}
Rational Number: Q = {a/b | a ∈ Z, b ∈ Z and b ≠ 0}
Real Number: R = {r | -∞<r<∞}
Complex Numbers: C = {z | z = a + bi, -∞<a<∞, -∞<b<∞}
Preparatory Programme - Mathematics
Subsets
A set “A” is said to be a subset of another set “B”, if all
the elements of set “A” are also elements of B, i.e. a ∈ A
⇒a∈B
Notation: A ⊆ B or “A is a subset of B”
Proper Subset: A set “A” is said to be a proper subset of
another set “B”, if all the elements of set “A” are also
elements of “B”, and in addition there exists at least one
element in “B” that is not in “A”
Notation: A ⊂ B or “A is a proper subset of B”
Preparatory Programme - Mathematics
Subsets
Examples:
A = {1, 2, 3}
B = {1, 2, 3, 4, 5}
A ⊆ B (also A ⊂ B)
A = N0
B=Z
A ⊆ B (also A ⊂ B)
A = {1, 2, 3}
A = N0
B = {1, 2, 3, 4, 5}
B = Z+
A ⊆ B (also A ⊂ B)
A ⊄ B (not a subset)
If A ⊆ B one can also write that B is a superset of A (B ⊇ A)
If A ⊂ B one can also write that B is a proper superset of A (B ⊃ A)
If A ⊄ B one can also write that B is not a superset of A (B ⊅ A)
Preparatory Programme - Mathematics
Set Equality
If two sets contain exactly the same elements then they
are said to be equal
Examples:
A = {cat, dog, mouse} B = {dog, mouse, cat}
A=B
A = {cat, dog, mouse} B = {dog, mouse, cat, rat} A ≠ B
If A ⊆ B and B ⊆ A, then it implies that A = B
Is this possible, A ⊂ B and B ⊂ A?
Preparatory Programme - Mathematics
Cardinality
If a set contains “n” distinct countable elements then the
cardinality of the set is said to be “n”
Example:
A = {laptop, smartphone, tablet}
|A| = 3
B = {{pigeon, sparrow}, {lion, tiger}, rat}
C = {x | x > 1}
|B| = 3
|C| = Infinite
D=∅
|D| = 0
Preparatory Programme - Mathematics
Power Set
The power set of a set “A” is another set “P(A)” that
contains all the possible subsets of “A”
Example:
A = {a, b, c}
P(A) = {∅, a, b, c, {a, b}, {a, c}, {b, c}, {a, b, c}}
If cardinality of A is n, i.e. |A| = n, then the cardinality
of the power set P(A) is 2n, i.e. |P(A)| = 2|A|
How do we get this? We will get to know while
studying combinatorics.
Preparatory Programme - Mathematics
Ordered n-tuple
The ordered n-tuple (x1, x2, x3, …, xn) is a sequence or
ordered collection of “n” objects.
Note:
1. 
A tuple may contain multiple instances of objects, i.e (5, 2, 2, 3) ≠ (5, 2, 3)
2. 
A tuple always contains finite elements
3. 
For sets, {A, B, C} = {C, B, A}, but for tuples, (A, B, C) ≠ (C, B, A)
Preparatory Programme - Mathematics
Cartesian Product
The Cartesian product of two sets “A” and “B” is defined as:
A×B = {(a, b) | a ∈ A, b ∈ B}
Example:
A = {a1, a2, a3} and B = {b1, b2}
Then A×B = {(a1, b1), (a1, b2), (a2, b1), (a2, b2), (a3, b1), (a3, b3)}
Note: A×B ≠ B×A
Preparatory Programme - Mathematics
Set Union
“A union B” is the set of all elements that
are in A, or B, or both.
Notation: A ∪ B
Preparatory Programme - Mathematics
Set Intersection
“A intersection B” is the set of all
elements that are present in both A and B
Notation: A ∩ B
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Set Complement
“A complement,” or “not A” is the set of
all elements that are not present in A
Notation: Ac or A’
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Set Theoretic Difference
The set theoretic difference of sets B and
A is the set of elements in B but not in A
Notation:
B – A or B \ A = {x ∈ B | x ∉ A}
Preparatory Programme - Mathematics
Venn Diagram: Intersection
Ω
A
B
A∩B
Preparatory Programme - Mathematics
Venn Diagram: Union
Ω
A
B
A∪B
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Venn Diagram: Complement
Ω
A
A’
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Venn Diagram: Universal Set
A
B
Ω
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Venn Diagram: Subset
A B
A⊆B
Preparatory Programme - Mathematics
Venn Diagram: Examples
Ω
A
B
(A ∪ B)’
Preparatory Programme - Mathematics
Venn Diagram: Examples
Ω
A
B
A’ ∩ B’
Preparatory Programme - Mathematics
Venn Diagram: Examples
Ω
A
B
(A ∪ B)’ = A’ ∩ B’
Preparatory Programme - Mathematics
Venn Diagram: Examples
Ω
A
B
(A ∩ B)’
Preparatory Programme - Mathematics
Venn Diagram: Examples
Ω
A
B
A’ ∪ B’
Preparatory Programme - Mathematics
Venn Diagram: Examples
Ω
A
B
(A ∩ B)’ = A’ ∪ B’
Preparatory Programme - Mathematics
Venn Diagram: Examples
Ω
A
B
A ∪ B’
Preparatory Programme - Mathematics
Venn Diagram: Examples
C
Ω
B
A
A∩B∩C
Preparatory Programme - Mathematics
Venn Diagram: Examples
C
Ω
B
A
(A ∪ B ∪ C)’
Preparatory Programme - Mathematics
Set Theory Rules
• 
• 
• 
• 
• 
Commutative Laws:
A∪B=B∪A
A∩B=B∩A
Associative Laws:
(A ∪ B) ∪ C = A ∪ (B ∪ C )
(A ∩ B) ∩ C = A ∩ (B ∩ C)
Distributive Laws:
(A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C)
(A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C)
DeMorgan’s Laws:
(A ∩ B)’ = (A)’ ∪ (B)’
(A ∪ B)’ = (A)’ ∩ (B)’
|A ∪ B| = |A| + |B| - |A ∩ B|
Preparatory Programme - Mathematics
Question
In a population of 100 people, 10 are rich, 5 are famous, and 3 are both
rich and famous. How many persons are:
= 90
•  not rich?
=7
•  rich but not famous?
=9
•  either rich or famous?
Preparatory Programme - Mathematics
Question
What is the number of elements in the power set of the set
{1, 2, 2, {1,2}}? Write down that power set.
If A is a subset of B, and B is a proper subset of the sample
space, then AC cannot be a subset of B. (True/False)
Preparatory Programme - Mathematics
Question
In a class of 120 students numbered 1 to 120, all even
numbered students opt for Physics, whose numbers
are divisible by 5 opt for Chemistry and those whose
numbers are divisible by 7 opt for Math. How many opt
for none of the three subjects?
Tip: |A∪B∪C| = |A| + |B| + |C| - |A∩B| - |B∩C| - |C∩A| + |A∩B∩C|
Solution: A denotes Physics, B denotes Chemistry, C denotes Math
|A| = 60
|B| = 24
|C| = 17
|A∩B| = 12 |B∩C| = 3
|C∩A| = 8
|A∩B∩C| = 1
|A∪B∪C| = 79
Answer = 120 – 79 = 41
Preparatory Programme - Mathematics
Question
The schedule of G first year students was inspected. It was found
that M were taking a Mathematics course, L were taking a
Language course and B were taking both a Mathematics course
and a Language course. What is the percentage of the students
whose schedule was inspected who were taking neither a
mathematics course nor a language course?
Solution:
Students taking at least one of the two courses = |M U L| = M+L–B
Number of students taking neither of the two courses = G-(M+L–B)
Expressing it in percentage: G-(M+L–B)/G×100
Preparatory Programme - Mathematics
Thank you
Preparatory Programme - Mathematics