What is a Set?

In Maths, sets are a collection of well-defined objects or elements. A set is represented by a capital letter symbol and the number of elements in the finite set is represented as the cardinal number of a set in a curly bracket {..}. For example, set A is a collection of all the natural numbers, such as A = { 1,2,3,4,5,6,7,8,…..∞}.

Sets can be represented in three forms:

  1. Roster Form: Example- Set of even numbers less than 8={2,4,6}
  2. Statement Form: Example-A = {Set of Odd numbers less than 9}
  3. Set Builder Form: Example: A = {x: x=2n, n ∈ N and 1 ≤ n ≤ 4}

What are the types of Sets?

A set has many types, such as;

  1. Empty Set or Null set: It has no element present in it.

    Example: A = { } is a null set.

  2. Finite Set: It has a limited number of elements.

    Example: A = { 1,2,3,4}

  3. Infinite Set: It has an infinite number of elements.

    Example: A = {x: x is the set of all whole numbers}

  4. Equal Set: Two sets which have the same members.

    Example: A = { 1,2,5} and B={2,5,1}: Set A = Set B

  5. Subsets: A set ‘A’ is said to be a subset of B if each element of A is also an element of B.

    Example: A={1,2}, B={1,2,3,4}, then A⊆B

  6. Universal Set: A set which consists of all elements of other sets present in a Venn diagram.

    Example: A={1,2}, B={2,3}, The universal set here will be, U = {1, 2,3}

Formulas For Sets

Let us consider A, B and C are three sets.

Formulas on sets

Questions and Solutions on Sets

  1. What is a Power set? Define with example.
  2. Answer: In set theory, the power set of a set A is defined as the set of all subsets of the Set A including the Set itself and the null or empty set.

    For Example: A = { 1, 3, 5}

    Power set of A, P(A) = {{ },{1},{3},{5},{1,3},{3,5},{1,5},{1,3,5}}

  1. Let A and B be two finite sets such that n(A) = 25, n(B) = 20 and n(A ∪ B) = 30, find n(A ∩ B).
  2. Solution: By the formula n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

    Hence, n(A ∩ B) = n(A) + n(B) – n(A ∪ B)

    = 25 + 20 – 30

    = 45 – 30

    = 15

  1. If n(A – B) = 30, n(A ∪ B) = 65 and n(A ∩ B) = 22, then find n(B).
  2. Solution: By the formula, we know;

    n(A∪B) = n(A – B) + n(A ∩ B) + n(B – A)

    65 = 30 + 22 + n(B – A)

    65 = 52 + n(B – A)

    n(B – A) = 65 – 52

    n(B – A) = 7

    Now n(B) = n(A ∩ B) + n(B – A)

    = 22 + 7

    = 15

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