Set Theory

Preparatory Programme - Mathematics

Set Theory

Ankur Sinha, PhD Production and Quantitative Methods Indian Institute of Management Ahmedabad

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

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

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 (asinha@iimahd.ernet.in) only for

queries about the course practicalities and not for discussing doubts or solutions to problems

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

Notations

A = {a1, a2, a3, a4, a5} B = {b1, b2, b3, b4} 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

Finite Sets

Special Sets

Null Set: A set with no elements Denoted as ∅ Universal Set: A set containing all elements under consideration Denoted as Ω

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, …} 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<∞}

Infinite but countable sets

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”

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} B = {1, 2, 3, 4, 5} A ⊆ B (also A ⊂ B) A = N0 B = Z+ 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)

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?

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} |B| = 3 C = {x | x > 1} |C| = Infinite D = ∅ |D| = 0

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.

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)

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

Set Union

“A union B” is the set of all elements that are in A, or B, or both. Notation: A ∪ B

Set Intersection

“A intersection B” is the set of all elements that are present in both A and B Notation: A ∩ B

Set Complement

“A complement,” or “not A” is the set of all elements that are not present in A Notation: Ac or A’

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}

Venn Diagram: Intersection

A ∩ B

Venn Diagram: Union

A ∪ B

Venn Diagram: Complement

Venn Diagram: Universal Set

Venn Diagram: Subset

A ⊆ B

Venn Diagram: Examples

(A ∪ B)’

A’ ∩ B’

(A ∪ B)’ = A’ ∩ B’

(A ∩ B)’

A’ ∪ B’

(A ∩ B)’ = A’ ∪ B’

A ∪ B’

A ∩ B ∩ C

(A ∪ B ∪ C)’

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|

In a population of 100 people, 10 are rich, 5 are famous, and 3 are both rich and famous. How many persons are: •  not rich? •  rich but not famous? •  either rich or famous?

Question

= 90 = 7 = 9

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)

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

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

Thank you

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