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:

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:

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:

1. Roster Form or Tabular form
2. 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

Set Theory Symbols

There are several symbols that are adopted for common sets. They are given in the table below:

Table 1: Symbols denoting common sets

Other Notations

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:

• Union of sets
• Intersection of sets
• Complement of sets
• Difference of sets

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}.

Video Lessons

Introduction to Sets

Principle of Inclusion and Exclusion

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