Sets

Key Concepts

• A set is a well-defined collection of objects.
• Sets can be represented in two ways — Roster or Tabular form and Set Builder form.
• Roster form: All the elements of a set are listed and separated by commas and are enclosed within braces {   }. Elements are not repeated generally.
• Set Builder form: In set builder form, a set is denoted by stating the properties which its members satisfy.
• A set does not change if one or more elements of the set are repeated.
• An empty set is the set having no elements in it. It is denoted by φ or { }.
• A set having a single element is called a singleton set.
• On the basis of the number of elements, sets are of two types — finite and infinite sets.
• A finite set is a set in which there are a definite number of elements. Now, φ or { } or null set is a finite set as it has 0 number of elements, which is a definite number.
• A set which is not finite is called an infinite set.
• All infinite sets cannot be described in the roster form.
• Two sets are equal if they have exactly the same elements.
• Two sets are said to be equivalent if they have the same number of elements.
• Set A is a subset of set B if every element of A is in B, i.e. there is no element in A which is not in B and is denoted by A ⊆ B.
• A is a proper subset of B if and only if every element in A is also in B and there exists at least one element in B that is not in A.It is denoted by A ⊂ B.
• If A is a proper subset of B, then B is a superset of A and is denoted by B ⊇ A.
• Let A be a set. Therefore, the collection of all subsets of A is called the power set of A and is denoted by P(A).
• Common set notations
  • N: The set of all natural numbers
  • Z: The set of all integers
  • Q: The set of all rational numbers
  • R: The set of real numbers
  • Z⁺: The set of positive integers
  • Q⁺: The set of positive rational numbers
  • R⁺: The set of positive real numbers
  • N ⊂ R , Q ⊂ R , Q ⊂ Z, R ⊂ Z and N ⊂ R+
• If A ⊆ B and B ⊆ A, then A = B.
• Null set φ is the subset of every set including the null set itself.
• Open interval: The interval which contains all the elements between a and b excluding a and b.
  In set notations: (a, b) = { x: a < x < b}
  
  Closed interval: The interval which contains all the elements between a and b and also the end points a and b is called the closed interval.
  [a, b] = {x: a ≤ x ≤ b}
  
• Semi-open intervals
  [a, b) = {x: a ≤ x < b} includes all the elements from a to b including a and excluding b.
  
  (a, b] = {x: a < x ≤ b} includes all the elements from a to b excluding a and including b.
  
• Universal set refers to a particular context.
  It is the basic set which is relevant to that context. The universal set is usually denoted by U.
• Union of sets A and B, denoted by A ∪ B is defined as the set of all the elements which are either in A or B or both.
• Intersection of sets A and B, denoted by A ∩ B, is defined as the set of all the elements which are common to both A and B.
• The difference of the sets A and B is the set of elements, which belong to A but not to B and it is written as A – B and read as ‘A minus B’.
  In set notations A – B = {x: x ∈ A, x ∉ B} and B – A = {x: x ∈ B, x ∉ A}.
• If the intersection of two non-empty sets is empty, i.e. A ∩ B = φ, then A and B are disjoint sets.
• Let U be the universal set and A be a subset of U. Thus, the complement of A, written as A’ or A^c, is the set of all elements of U which are not in set A.
• The number of elements present in a set is known as the cardinal number of the set or cardinality of the set. It is denoted by n(A).
• If A is a subset of U, then A’ is also a subset of U.
• Counting theorems are together known as the Inclusion–Exclusion Principle. This principle helps in determining the cardinality of union and intersection of sets.
• Sets can be represented graphically using Venn diagrams. Venn diagrams consist of rectangles and closed curves, usually circles. The universal set is generally represented by a rectangle and its subsets by circles.
  

Key Formulae

• Union of sets
  A ∪ B = {x: x ∈ A or x ∈ B } ∪∩∈∉
  
• Intersection of sets
  A ∩ B = {x: x ∈ A and x ∈ B }
  
• Complement of a set
  A’ = {x: x ∈ U and x ∉ A}, A’ = U – A
  
• Difference of sets
  A – B = {x: x ∈ A, x ∉ B} and B – A = {x: x ∈ B, x ∉ A}
  
• Symmetric difference of two sets:
  Let A and B be two sets.
  Hence, the symmetric difference of sets A and B is the set,
  (A – B) ∪ (B – A) and it is denoted by A △ B.
  A △ B = (A – B) ∪ (B – A) =  {x : x ∈ A,x ∈ B,x ∉ A ∩ B}
  
• Properties of the operation of union
  • Commutative law:
    A ∪ B = B ∪ A
  • Associative law:
    (A ∪ B) ∪ C = A ∪ (B ∪ C)
  • Law of identity:
    A ∪ φ = A
  • Idempotent law:
    A ∪ A = A
  • Law of U:
    U ∪ A = U
• Properties of operation of intersection
  • Commutative law:
    A ∩ B = B ∩ A
  • Associative law:
    (A ∩ B) ∩ C = A ∩ (B ∩ C)
  • Law of φ and U:
    φ  ∩ A = φ and U ∩ A = U
  • Idempotent law:
    A ∩ A = A
  • Distributive law:
    A ∩ (B ∩ C) = (A ∩ B) ∪ (A ∩ C)
• Properties of complement of sets
  • Complement laws:
    A ∪ A’ = U
    A ∩ A’ = φ
  • De-Morgan’s law:
    (A ∪ B)’ = A’ ∩ B’
    (A ∩ B)’ = A’ ∪ B’
  • Law of double complementation:
    (A’)’ = A
  • Laws of empty set and universal set:
    φ’ = U and U’ = φ
• Operations on sets
  • A – B = A ∩ B'
  • B – A = A' ∩ B
  • A – B = A ⇔ A ∩ B = φ
  • (A – B) ∪ B = A ∪ B
  • (A – B) ∩ B = φ
  • A ⊆ B ⇔ B' ⊆ A'
  • (A – B) ∪ (B – A) = (A ∪ B) – (A ∪ B)
• Some more important results:
  Let A, B and C be three sets. Hence,
  • A – (B ∩ C) = (A – B) ∪ (A – C)
  • A – (B ∪ C) = (A – B) ∩ (A – C)
  • A ∩ (B – C) = (A ∩ B) – (A ∩ C)
  • A ∩ (B △ C) = (A – B) △ (A – C)
• Counting Theorems
  • If A and B are finite sets, then the number of elements in the union of two sets is given by
    n(A ∪ B) = n(A) + n(B) – n(A ∪ B)
  • If A and B are finite sets and A ∩ B = φ, then
    n(A ∪ B ) = n(A) + n(B)
  • n(A ∪ B) = n(A – B) + n(B – A) + n(A ∩ B)
  • n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(B ∩ C) – n(A ∩ B) - n(A ∩ C) + n(A ∩ B ∩ C)
  • Number of elements in exactly two of the sets =
    n(A ∩ B) + n(B ∩ C) + n(C ∩ A) – 3n(A ∩ B ∩ C)
  • Number of elements in exactly one of the sets =
    n(A) + n(B) + n(C) – 2n(A ∩ B) – 2n(B ∩ C) – 2n(C ∩ A) + 3n(A ∩ B ∩ C)
  • n(A' ∪ B') = n((A ∩ B)') = n(U) – n(A ∩ B)
  • n (A' ∩ B') = n((A ∪ B)') = n(U) – n(A ∪ B)
• Number of elements in the power set of a set with n elements = 2ⁿ
  Number of proper subsets in the power set = 2ⁿ – 1