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What is a Set?
A set is a well-defined collection of distinct objects.
Example: A = {1, 2, 3, 4, 5}

What is an element of a Set?

  • The objects in a set are called its elements.
    • So in case of the above Set A, the elements would be 1, 2, 3, 4, and 5. We can say, 1 ∈ A, 2 ∈ A
  • Usually we denote Sets by CAPITAL LETTERs like A, B, C, etc. while their elements are denoted in small letters like x, y, z
  • If x is an element of A, then we say x belongs to A and we represent it as x ∈ A
  • If x is not an element of A, then we say that x does not belong to A and we represent it as x ∉ A

How to describe a Set?

  • Roaster Method or Tabular Form
    • In this form, we just list the elements
    • Example A = {1, 2, 3, 4} or B = {a, b, c, d, e}
  • Set- Builder Form or Rule Method or Description Method
    • In this method, we list the properties satisfied by all elements of the set

Example A = {x : x ∈ N, x< 5}

Some examples of Roster Form vs Set-builder Form

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Sets of Numbers

  1. Natural Numbers (N)

N = {1, 2, 3,4 ,5 6, 7, …}

  1. Integers (Z)

Z = {…, -3, -2, -1, 0, 1, 2, 3, 4, …}

  1. Whole Numbers ( W )

W = {0, 1, 2, 3 4, 5, 6…}

  1. Rational Numbers (Q)

{p/q : p ∈ Z, q ∈ Z, q ≠0}

Finite Sets & Infinite Sets

Finite Set: A set where the process of counting the elements of the set would surely come to an end is called finite set

  • Example: All natural numbers less than 50
  • All factors of the number 36

Infinite Set: A set that consists of uncountable number of distinct elements is called infinite set.

Example: Set containing all natural numbers {x | x ∈ N, x > 100}

Cardinal number of Finite Set

The number of distinct elements contained in a finite set A is called the cardinal number of A and is denoted by n(A)

  • Example A = {1, 2, 3, 4} then n(A) = 4
  • A = {x | x is a letter in the word ‘APPLE’}. Therefore A = {A, P, L, E} and n(A) = 4
  • A = {x | x is the factor of 36}, Therefore A = { 1, 2, 3, 4, 6, 9, 12, 18, 36} and n(A) = 9

Empty Set

  • A set containing no elements at all is called an empty set or a null set or a void set.
  • It is denoted by ϕ (phai)
  • In roster form you write ϕ = { }
  • Also n (ϕ) = 0
    • Examples: {x | x ∈ N, 3 <  x < 4} = ϕ 
    • {x | x is an even prime number, x > 5} = ϕ

Non Empty Set

  • A set which has at least one element is called a non-empty set
    • Example: A = {1, 2, 3} or B = {1}

Singleton Set

  • A set containing exactly one element is called a singleton set
    • Example: A = {a} or B = {1}

Equal Sets

  • Two set A and B are said to be equal sets and written as A = B if every element of A is in B and every element of B is in A
    • Example A = {1, 2, 3, 4} and B = {4, 2, 3, 1}
  • It is not about the number of elements. It is the elements themselves.
  • If the sets are not equal, then we write as A ≠ B

Equivalent Sets

  • Two finite sets A and B are said to be equivalent, written as A ↔ B, if n(A) = n(B), that is they have the same number of elements.
    • Example: A = {a, e, i, o, u} and B = {1, 2, 3, 4, 5}, Therefore n(A) = 5 and n(B) = 5 therefore A ↔ B
  • Note: Two equal sets are always equivalent but two equivalent sets need not be equal.

Subsets

  • If A and B are two sets given in such a way that every element of A is in B, then we say A is a subset of B and we write it as A ⊆ B
  • Therefore is A ⊆ B and x ∈ A then x ∈ B 
  • If A is a subset of B, we say B is a super set of A and is written as B ⊇ A
  • Every set is a subset of itself.
    • i.e. A ⊆ A, B ⊆ B etc.
  • Empty set is a subset of every set
    • i.e. ϕ ⊆ A, ϕ ⊆ B
  • If A ⊆ B and B ⊆ A, then A = B
  • Similarly, if A = B, then A ⊆ B and B ⊆ A
  • If set A contains n elements, then there are 2n subsets of A

Power Set

  • The set of all possible subsets of a set A is called the power set of A, denoted by P(A). If A contains n elements, then P(A) = 2n
    • i.e. if A = {1, 2}, then P(A) = 22 = 4
    • Empty set is a subset of every set
    • So in this case the subsets are {1}, {2}, {2, 3} & ϕ

Proper Subset

Let A be any set and let B be any non-empty subset. Then A is called a proper subset of B, and is written as A ⊂ B  , if and only if every element of A is in B, and there exists at least one element in B which is not there in A.

  • i.e. if A ⊆ B and  A ≠ B then A⊂ B
  • Please note that ϕ has no proper subset
  • A set containing n elements has (2n – 1) proper subsets.
    • i.e. if A = {1, 2, 3, 4}, then the number of proper subsets is (24 – 1) = 15

Universal Set

  • If there are some sets in consideration, then there happens to be a set which is a super set of each one of the given sets. Such a set is known as universal set, to be denoted by U or ξ .
    • i.e. if A = {1, 2}, B = {3, 4}, and C = {1, 5}, then U or ξ = {1, 2, 3, 4, 5}

Operations on Sets

Union of Sets

  • The union of sets A and B, denoted by A∪ B  , is the set of all those elements, each one of which is either in A or in B or in both A and B
  • If there is a set A = {2, 3} and B = {a, b}, then A∪ B = {2, 3, a, b}
  • So if A∪ B = {x | x ∈ A or x ∈ B}  then  x ∈ A ∪ B  which means x ∈ A or x ∈ B
  • And if x  ∉ A ∪ B  which means x ∉ A or x ∉ B  

Interaction of Sets

  • The intersection of sets A and B is denoted by A ∩ B  , and is a set of all elements that are common in sets A and B.
  • e. if A = {1, 2, 3} and B = {2, 4, 5}, then A ∩ B = {2}  as 2 is the only common element.
  • Thus A ∩ B = {x : x ∈ A and x ∈ B} then x ∈ A ∩ B i.e. x ∈ A and x ∈ B 
  • And if x ∉ A ∩ B  i.e. x ∉ A and x ∉ B 

Disjointed Sets

  • Two sets A and B are called disjointed, if they have no element in common. Therefore:
  • x ∉ A ∩ B  i.e. x ∉ A and x ∉ B

Intersecting sets

  • Two sets are said to be intersecting or overlapping or joint sets, if they have at least one element in common.
  • Therefore two sets A and B are overlapping if and only if A ∩ B ≠ ϕ 
  • Intersection of sets is Commutative
    • i.e. A ∩ B  = B ∩ A for any sets A and B
  • Intersection of sets is Associative i.e. for any sets, A, B,  C,
    • (A ∩ B) ∩ C = A ∩ ( B ∩ C)
    • A ⊆ B, then A ∩ B = A
      • Since A ⊆  ξ, so A ∩ ξ = A
    • For any sets A and B we have
      •  A ∩ B ⊆ A and A ∩ B ⊆ B
      • A ∩  ϕ = ϕ for every set A

 

Difference of Sets

  • For any two sets A and B, the difference A – B is a set of all those elements of A which are not in B.

i.e. if A = {1, 2, 3, 4, 5} and B = {4, 5, 6}, Then  A – B = {1, 2, 3} and B – A = {6}

Therefore A – B = { x | x ∈ A and x ∉ B}, then x ∈ A – B then x ∈ A but x ∉ B

  • If A ⊆ B then A – B = ϕ

 

Complement of a Set

Let x be the universal set and let A ⊆ x. The the complement of A, denoted by A’ is the set if all those elements of x which are not in A.

  • i.e. let ξ = {1, 2, 3, 4, 5, 6, 7, 8} and A= {2, 3, 4}, then A’={1, 5, 6, 7, 8}
  • Thus A’={x | x ∈ ξ and x ∉ A} clearly x ∈A’ and x ∉ A

Please note

  • ϕ’ =  ξ and ξ’ = ϕ
  • A ∪ A’ = ξ and A ∩ A’ = ϕ

 

Disruptive laws for Union and Intersection of Sets

For any three sets , B, C, we have the following

  •   A ∪ ( B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Say A = {1,2}, B = {2, 3} and C = {3, 4}

Therefore  A ∪ ( B ∩ C) = {1, 2, 3} and (A ∪ B) ∩ (A ∪ C) = {1, 2, 3} and hence equal

  • A ∩ ( B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Say A = {1,2}, B = {2, 3} and C = {3, 4}

Then A ∩ ( B ∪ C) = {2} and (A ∩ B) ∪ (A ∩ C) = {2} and hence equal

 

Disruptive laws for Union and Intersection of Sets

De-Morgan’s Laws

Let A and B be two subsets of a universal set ξ, then

  • (A ∪ B)’ = A’ ∩ B’
  • (A ∩ B)’ = A’∪ B’

Let ξ = {1,2,3,4,5,6} and A = {1,2,3} and B = {3,4,5}

Then A ∪ B = { 1, 2, 3, 4, 5}, therefore (A ∪ B)’ = {6}

A’ = {4, 5, 6} and B’={1, 2, 6}

Therefore A’ ∩ B’ = {6}. Hence proven.

 

 

 

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