basic of set theory

Basic Concept of Set Theory (Data Interpretation)

SET THEORY :- The process of collecting items, or sets, known as elements or numbers, is known as set theory. Every object in mathematics is thought to be a set, and every type of theorem is thought to be predicate calculus. It was taken by the axioms of Set Theory. Let’s understand what are the set theory formulas which are important to understand.

Set Theory is an area of mathematics that teaches us about sets and their properties. A set is a group of objects or a collection of objects. These items are frequently referred to as set elements or members. A set of cricket players, for example, is a group of players.

In other word we can say that set is a collections of well defined elements . i.e. well defined means object of the set follow some certain rule of the set.

Sets Representation

There are two ways to express sets:

  1. Tabular or roster format

Set  Builder formate

Form of Roster

All of the set’s elements are listed in roster form, separated by commas and wrapped 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. 

Form for the Builder

All elements in the set builder form share a common property. This property does not apply to items that are not part of 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.

 

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 Notation 

Before we go on and explore various types of set theory formulas, let us first understand how we can denote a set. The common way of denoting a set is listing it within curly brackets. For Example: A= {yellow, green, pink, blue, red} is a set of colours, W= {0,1, 2,3,4… 15} a set of whole numbers upto 15. 

Defining a Set 

Whilst writing a set, it is mandatory to define or explain what asset is composed of. Mentioning a proper description of the set gives a clear idea about its members. For Example: 

  • B= {a, e, i, o, u} is a set containing 5 elements that are are vowels of the English alphabets 
  • H= {-10…-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5…} is a set of of integers

Equal Sets  

Two given sets X and y I will only be equal if both have the exact and same number of elements. For example: X= {8, 7, 9, 10} and Y= {7, 9, 8, 10} 

Thus, X= Y 

Null Set 

An empty set for a null set is a vital concept of the set theory. If there is a set without any elements in it, we can call it a null set. An empty set is denoted by A {} or A Φ.

Union of Sets 

Union of two sets can be defined as A or B, which means that there is a new set that contains the elements that appear in set A as well as set B. For Example: 

C= {6,7,8,5,9} 

D= {5, 6,4, 3,1,2} 

C ∪ D= {1,2,3,4,5,6,7,8,9}

Intersection of Sets

You will come across a variety of questions based on intersection of sets which you will be able to solve using the set theory formulas. The intersection of two given sets is the number of elements that are common to both sets. For example: 

E= {6,7,4,5,3,1}

F= {1,2,3,4,5}

E∩ F ={1,3,4,5}

Set Theory Formulas on Properties

  • Commutativity:
    • A⋂B = B⋂A 
    • A∪B = B∪A
  • Associativity:
    • A⋂ (B⋂C) = (A⋂B)⋂C
    • A∪ (B∪C) = (A∪B)∪C
  • Distributivity: A ⋂(B∪C) = (A ⋂B) ∪ (A⋂C)
  • Idempotent Law:
    • A ⋂ A = A
    • A ∪ A = A
  • Law of Ø and ∪:
    • A ⋂ Ø = Ø
    • U ⋂ A = A
    • A ∪ Ø = A
    • U ∪ A = U

Sets Theory Formulas of Difference of Sets

  • A – A = Ø
  • B – A = B⋂ A’
  • B – A = B – (A⋂B)
  • n(AUB) = 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)
  • (A – B) = A if A⋂B =  Ø
  • (A – B) ⋂ C = (A⋂ C) – (B⋂C)
  • A ΔB = (A-B) U (B- A) 
  • n(A) = n(∪) – n(A)

Sets Theory Formulas of Complement Sets

  • Law of Double complementation: (A’)’ = A
  • Laws of Empty set and Universal Set: Ø’ = ∪ and ∪’ = Ø
  • Complement Law : A∪A’ = U, A⋂A’ = Ø and A’ = U –  A
  • De Morgan’s Laws: (A ∪B)’ = A’ ⋂B’ and (A⋂B)’ = A’ ∪ B’

for details see video click on 

For Practices Click on below link  

 

 

 

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