Set Theory Mathematical Economics

Mathematical Economics

Mathematical economics is the application of mathematical method to represent theories and analysed problem in Economics.
Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific positive claims about controversial or contentious subjects that would be impossible without mathematics.

Today we will discuss about Set theory

Set Theory is a branch of mathematical logic where we learn sets and their properties. A set is a collection of objects or groups of objects. These objects are often called elements or members of a set. For example, a group of players in a cricket team is a set.

Since the number of players in a cricket team could be only 11 at a time, thus we can say, this set is a finite set. Another example of a finite set is a set of English vowels. But there are many sets that have infinite members such as a set of natural numbers, a set of whole numbers, set of real numbers, set of imaginary numbers, etc.

Set Theory Origin

Georg Cantor (1845-1918), a German mathematician, initiated the concept ‘Theory of sets’ or ‘Set Theory’. While working on “Problems on Trigonometric Series”, he encountered sets, that have become one of the most fundamental concepts in mathematics. Without understanding sets, it will be difficult to explain the other concepts such as relations, functions, sequences, probability, geometry, etc.

Definition of Sets

As we have already learned in the introduction, set is a well-defined collection of objects or people. Sets can be related to many real-life examples, such as the number of rivers in India, number of colours in a rainbow, etc.

Example

To understand sets, consider a practical scenario. While going to school from home, Nivy decided to note down the names of restaurants which come in between. The list of the restaurants, in the order they came, was:

$L i s t 1 : R_{A} R_{B} R_{C} R_{D} R_{E}$

The above-mentioned list is a collection of objects. Also, it is well-defined. By well-defined, it is meant that anyone should be able to tell whether the object belongs to the particular collection or not. E. g. a stationary shop can’t come in the category of the restaurants. If the collection of objects is well-defined, it is known as a set.

The objects in a set are referred to as elements of the set. A set can have finite or infinite elements. While coming back from the school, Nivy wanted to confirm the list what she had made earlier. This time again, she wrote the list in the order in which restaurants came. The new list was:

$L i s t 2 : R_{E} R_{D} R_{C} R_{B} R_{A}$

Now, this is a different list. But is a different set? The answer is no. The order of elements has no significance in sets so it is still the same set.

Representation of Sets

Sets can be represented in two ways:

Roster Form or Tabular form
Set Builder Form

Roster Form

In roster form, all the elements of the set are listed, separated by commas and enclosed between curly braces { }.

Example: If set represents all the leap years between the year 1995 and 2015, then it would be described using Roster form as:

A ={1996,2000,2004,2008,2012}

Now, the elements inside the braces are written in ascending order. This could be descending order or any random order. As discussed before, the order doesn’t matter for a set represented in the Roster Form.

Also, multiplicity is ignored while representing the sets. E.g. If $L$ represents a set that contains all the letters in the word ADDRESS, the proper Roster form representation would be

L ={A,D,R,E,S }= {S,E,D,A,R}

L≠ {A,D,D,R,E,S,S}

Set Builder Form

In set builder form, all the elements have a common property. This property is not applicable to the objects that do not belong to the set.

Example: If set S has all the elements which are even prime numbers, it is represented as:

S={ x: x is an even prime number}

where ‘x’ is a symbolic representation that is used to describe the element.

‘:’ means ‘such that’

‘{}’ means ‘the set of all’

So, S = { x:x is an even prime number } is read as ‘the set of all x such that x is an even prime number’. The roster form for this set S would be S = 2. This set contains only one element. Such sets are called singleton/unit sets.

Another Example:

F = {p: p is a set of two-digit perfect square numbers}

How?

F = {16, 25, 36, 49, 64, 81}

We can see, in the above example, 16 is a square of 4, 25 is square of 5, 36 is square of 6, 49 is square of 7, 64 is square of 8 and 81 is a square of 9}.

Even though, 4, 9, 121, etc., are also perfect squares, but they are not elements of the set F, because the it is limited to only two-digit perfect square.

Types of Sets

The sets are further categorised into different types, based on elements or types of elements. These different types of sets in basic set theory are:

Finite set: The number of elements is finite
Infinite set: The number of elements are infinite
Empty set: It has no elements
Singleton set: It has one only element
Equal set: Two sets are equal if they have same elements
Equivalent set: Two sets are equivalent if they have same number of elements
Power set: A set of every possible subset.
Universal set: Any set that contains all the sets under consideration.
Subset: When all the elements of set A belong to set B, then A is subset of B

Symbol	Corresponding Set
N	Represents the set of all Natural numbers i.e. all the positive integers. This can also be represented by $Z^{+}$ . Examples: 9, 13, 906, 607, etc.
Z	Represents the set of all integers The symbol is derived from the German word Zahl, which means number. Positive and negative integers are denoted by $Z^{+}$ and $Z^{-}$ respectively. Examples: -12, 0, 23045, etc.
Q	Represents the set of Rational numbers The symbol is derived from the word Quotient. It is defined as the quotient of two integers (with non-zero denominator) Positive and negative rational numbers are denoted by $Q^{+}$ and $Q^{-}$ respectively. Examples: $\frac{13}{9}$ , $- \frac{6}{7}$ , $\frac{14}{3}$ , etc.
R	Represents the Real numbers i.e. all the numbers located on the number line. Positive and negative real numbers are denoted by $R^{+}$ and $R^{-}$ respectively. Examples: $4.3$ , $π$ , $4 \sqrt{3}$ ,, etc.
C	Represents the set of Complex numbers. Examples: 4 + 3i, i, etc

Symbol	Symbol Name
{ }	set
A ∪ B	A union B
A ∩ B	A intersection B
A ⊆ B	A is subset of B
A ⊄ B	A is not subset B
A ⊂ B	proper subset / strict subset
A ⊃ B	proper superset / strict superset
A ⊇ B	superset
A ⊅ B	not superset
Ø	empty set
P (C)	power set
A = B	Equal set
Ac	Complement of A
a∈B	a element of B
x∉A	x not element of A

Venn diagrams

Venn diagrams are a useful way of visualizing relationships between sets.
When we draw Venn diagrams, sets are represented by circles. The inside of a circle represents all the elements that are members of that set and the outside of the circle represents all elements which are not members of that set.
The areas where circles overlap represents the set of elements which are in all of the sets corresponding to the overlapping circles. We call this area the intersection of the (overlapping) sets.
The sample space is represented by a rectangle which is drawn around all of the circles and we write the set name of the sample space (typically the letter $S$ ) in the left-hand corner. If we want to partition the sample set into multiple parts, we draw lines to divide the rectangle into the required number of parts.

Worked Example

In an Economics seminar group there are $32$ students. $19$ of these students say they enjoy studying Microeconomics, $17$ say they enjoy studying Macroeconomics, and $5$ say that they enjoy studying neither Microeconomics nor Macroeconomics. How many of the students in the seminar group enjoy studying both Microeconomics and Macroeconomics?

Solution

We can solve this problem by drawing a Venn diagram to represent the situation and adding each new piece of information to the diagram as we go.
The sample space is the set of all $32$ students in the seminar group. We will denote the sample space by the letter $S$ .
Denote the set of ( $19$ ) students who enjoy studying Microeconomics by $I$ and the set of ( $17$ ) students who enjoy studying Macroeconomics by $A$ . These sets are represented by circles. We want to known how many students there are in the intersection of these two circles (who enjoy studying both subjects).
The $5$ students who enjoy neither of the subjects go on the outside of the circles. We know the total in the Microeconomics circle needs to be $19$ but we can't put this in the circle because we don't know how many should go in the intersection and how many should go in the right hand part of the Microeconomics circle (those who enjoy studying only Microeconomics). We have the same problem for the Macroeconomics circle so our Venn diagram currently looks like this:

Ven Diagram

We know there are

32

students in the group, and if there are

5

students outside the circles then the other three sections must add up to

27

. We know there are

17

students who enjoy Macroeconomics, so the middle and left section must add up to

17

. This leaves

10

in the right-hand section of the

I

circle because

27 - 17 = 10

Since there are

19

students who enjoy Microeconomics, the two parts of the Microeconomics circle should add up to

19

so we can now find the number in the intersection by subtracting

10

from

19

19 - 10 = 9

There are $19$ students who enjoy Microeconomics, and $27 - 19 = 8$ , so the number who enjoy Macroeconomics but not enjoy Macroeconomics must be $8$ .

we check all four facts we were given, we can now see they are all true. In particular, there are $32$ students in the seminar group and $32$ in the venn diagram.
There are $9$ students who enjoy both Microeconomics and Macroeconomics.
Symbol

we check all four facts we were given, we can now see they are all true. In particular, there are $32$ students in the seminar group and $32$ in the venn diagram.
There are $9$ students who enjoy both Microeconomics and Macroeconomics.

A complement Ac Every element. Is not element of A set{6.7.8__)

2,€ A 2 is element of Set A and belong to A

- 4 does not belong to C and not belong to C

A contain B Every element. Of B is also an element of A
B is subset of A

A U C. A union C Evey element which is member Of A&C

A intersection C every element which is member of A& C

Set Theory Formulas

n( A ∪ B ) = n(A) +n(B) – n (A ∪ B)
n(A∪B)=n(A)+n(B) {when A and B are disjoint sets}
n(U)=n(A)+n(B)–n(A∩B)+n((A∪B)c)
n(A∪B)=n(A−B)+n(B−A)+n(A∩B)
n(A−B)=n(A∩B)−n(B)
n(A−B)=n(A)−n(A∩B)
n(Ac)=n(U)−n(A)
n(PUQUR)=n(P)+n(Q)+n(R)–n(P⋂Q)–n(Q⋂R)–n(R⋂P)+n(P⋂Q⋂R)

Set Operations

The four important set operations that are widely used are:

Fundamental Properties of Set operations:

Like addition and multiplication operation in algebra, the operations such as union and intersection in set theory obeys the properties of associativity and commutativity. Also, the intersection of sets distributes over the union of sets.

Sets are used to describe one of the most important concepts in mathematics i.e. functions. Everything that you observe around you, is achieved with mathematical models which are formulated, interpreted and solved by functions.

Problems and Solutions

Q.1: If U = {a, b, c, d, e, f}, A = {a, b, c}, B = {c, d, e, f}, C = {c, d, e}, find (A ∩ B) ∪ (A ∩ C).

Solution: A ∩ B = {a, b, c} ∩ {c, d, e, f}

A ∩ B = { c }

A ∩ C = { a, b, c } ∩ { c, d, e }

A ∩ C = { c }

∴ (A ∩ B) ∪ (A ∩ C) = { c }

Q.2: Give examples of finite sets.

Solution: The examples of finite sets are:

Set of months in a year

Set of days in a week

Set of natural numbers less than 20

Set of integers greater than -2 and less than 3

Q.3: If U = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, A = {3, 5, 7, 9, 11} and B = {7, 8, 9, 10, 11}, Then find (A – B)′.

Solution: A – B is a set of member which belong to A but do not belong to B

∴ A – B = {3, 5, 7, 9, 11} – {7, 8, 9, 10, 11}

A – B = {3, 5}

According to formula,

(A − B)′ = U – (A – B)

∴ (A − B)′ = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11} – {3, 5}

(A − B)′ = {2, 4, 6, 7, 8, 9, 10, 11}.

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