Types of Sets

Overview of Sets

• Definition: A set is a well-defined collection of objects or elements with a common feature.
• Notation: Sets are denoted using curly braces {}.
• Examples:
  • Set of even numbers
  • Integers from 1 to 100
  • Collection of prime numbers

Types of Sets

1. Singleton Set

• Definition: A set with exactly one element.
• Examples:
  • Set A = {1}
  • Set P = {a : a is an even prime number} = {2}

2. Empty Set

• Also Known As: Null set or Void set.
• Definition: A set with no elements.
• Examples:
  • Set A = {a: a is a number greater than 5 and less than 3}
  • Set B = {p: p are the students studying in class 7 and class 8}

3. Finite Set

• Definition: A set with a finite number of elements.
• Examples:
  • Set A = {a: a is a whole number less than 20}
  • Set B = {a, b, c, d, e}

4. Infinite Set

• Definition: A set with an infinite number of elements.
• Examples:
  • Set A = {a: a is an odd number}
  • Set B = {2, 4, 6, 8, 10, 12, 14, ...}

5. Equal Set

• Definition: Two sets with the same elements and number of elements.
• Examples:
  • Set A = {1, 2, 6, 5}
  • Set B = {2, 1, 5, 6} (A = B)

6. Equivalent Set

• Definition: Sets with the same number of elements, not necessarily the same elements.
• Examples:
  • Set A = {2, 3, 5, 7, 11}
  • Set B = {p, q, r, s, t}

7. Subset

• Definition: Set A is a subset of Set B if all elements of A are in B.
• Symbol: A ⊆ B
• Examples:
  • Set A = {33, 66, 99}
  • Set B = {22, 11, 33, 99, 66} (A ⊆ B)

8. Power Set

• Definition: Set of all subsets of a set A.
• Notation: P(A)
• Example:
  • For set A = {a, b, c}, P(A) = { {}, {a}, {b}, {c}, {a, b}, {b, c}, {c, a}, {a, b, c} }

9. Universal Set

• Definition: A set which contains all elements relevant to a particular discussion.
• Examples:
  • For Sets A = {a, b, c, d} and B = {1, 2}, U = {a, b, c, d, e, 1, 2}

10. Disjoint Sets

• Definition: Two sets having no elements in common.
• Examples:
  • Set A = {a, b, c, d}
  • Set B = {1, 2}

Solved Examples

Example 1: Universal Set in Venn Diagram

• Sets:
  • Set A = {1, 2, 3, 4, 5}
  • Set B = {1, 2, 5, 0}
  • Universal Set U = {0, 1, 2, 3, 4, 5, 6, 7}

Example 2: Equal and Equivalent Sets

• Sets:
  • Set A = {2, 4, 6, 8, 10}
  • Set B = {a, b, c, d, e}
  • Set C = {c: c ∈ N, c is even, c ≤ 10}
  • Set D = {1, 2, 5, 10}
  • Set E = {x, y, z}
• Results:
  • Equivalent Sets: A, B, C
  • Equal Sets: A, C

Example 3: Types of Sets

• Sets:
  • Set A = {a: a is divisible by 10}
  • Set B = {2, 4, 6}
  • Set C = {p}
  • Set D = {n, m, o, p}
  • Set E = {}
• Results:
  • Infinite Set: A
  • Finite Sets: B, D
  • Singleton Set: C
  • Null Set: E

Example 4: Subsets

• Sets:
  • Set P = {0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20}
  • Set A = {a, 1, 0, 2}
  • Set B = {0, 2, 4}
  • Set C = {1, 4, 6, 10}
  • Set D = {2, 20}
  • Set E = {18, 16, 2, 10}
• Results:
  • Subsets of P: B, D, E

FAQs

• Sets: Well-defined collections of objects.
• Subsets: Contain some elements of the given set.
• Types of Sets: Empty, Non-Empty, Finite, Infinite, Singleton, Equivalent, Subset, Superset, Power, Universal.
• Difference between ∅ and {}:
  • ∅ represents a null set.
  • {} can represent a singleton set with element ∅.