Determine the Number of Subsets of a Set

Introduction

• The focus of this lecture is on determining the number of subsets of a set.
• Key Question: If a set contains n elements, how many distinct subsets can be formed?

Special Cases & Examples

1. Empty Set:

   • Contains 0 elements.
   • Only one subset exists: the empty set itself.

2. Single Element Set (e.g., {a}):

   • Contains 1 element.
   • Subsets: {a}, empty set.
   • Total: 2 subsets.

3. Two Element Set (e.g., {a, b}):

   • Contains 2 elements.
   • Subsets: {a, b}, {a}, {b}, empty set.
   • Total: 4 subsets.

4. Three Element Set (e.g., {a, b, c}):

   • Contains 3 elements.
   • Subsets: {a, b, c}, {a, b}, {b, c}, {a, c}, {a}, {b}, {c}, empty set.
   • Total: 8 subsets.

Identifying the Pattern

• Observation: Compare number of elements to the number of subsets:
  • 2 elements → 4 subsets = 2²
  • 3 elements → 8 subsets = 2³
  • Pattern: Number of subsets = 2^n where n is the number of elements.

General Rule

• If a set contains n elements, the number of distinct subsets is 2^n.

Proper Subsets

• Definition: Every subset except the set itself is a proper subset.
• Rule: Number of distinct proper subsets = 2^n - 1.

Examples

Example A

• Set: {w, x, y, z}
  • Total Elements: 4
  • Subsets: 2^4 = 16
  • Proper Subsets: 16 - 1 = 15

Example B

• Set: {x ∈ natural numbers | 7 ≤ x ≤ 14}
  • Written as: {7, 8, 9, 10, 11, 12, 13, 14}
  • Total Elements: 8
  • Subsets: 2^8 = 256
  • Proper Subsets: 256 - 1 = 255