Equal Sets, Subsets, and Power Sets [ English ]

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1. Introduction

In set theory, relationships between sets are fundamental for understanding structure and hierarchy. Three key concepts in this context are:

• Equal Sets
• Subsets
• Power Sets

These concepts are widely used in logic, combinatorics, and computer science.

2. Equal Sets

Definition

Two sets are said to be equal if they contain exactly the same elements.

Representation

A = B

Example

A = {1, 2, 3}
B = {3, 2, 1}

✔ A = B because:

• Order does not matter
• All elements are the same

Key Points

• Order of elements is irrelevant
• Repetition does not affect sets
• Every element of A must be in B and vice versa

3. Subsets

Definition

A set A is a subset of set B if every element of A is also an element of B.

Representation

A ⊆ B

Example

A = {1, 2}
B = {1, 2, 3}

✔ A ⊆ B

Proper Subset

If A ⊆ B and A ≠ B, then A is a proper subset of B.

A ⊂ B

Example

A = {1, 2}
B = {1, 2, 3}

✔ A ⊂ B

Important Properties

• Every set is a subset of itself:

  A ⊆ A
  

• Empty set is a subset of every set:

  ∅ ⊆ A
  

4. Power Set

Definition

The power set of a set A is the set of all possible subsets of A.

Representation

P(A)

Example

Let:

A = {1, 2}

Subsets of A:

• {}
• {1}
• {2}
• {1, 2}

Power set:

P(A) = { {}, {1}, {2}, {1, 2} }

Example 2

Let:

A = {a, b}

P(A) = { {}, {a}, {b}, {a, b} }

5. Number of Elements in Power Set

If a set has n elements, then:

|P(A)| = 2^n

Example

A = {1, 2, 3}

Number of subsets:

|P(A)| = 2^3 = 8

6. Difference Between Concepts

7. Key Observations

• Order does not matter in sets
• Subsets define relationships between sets
• Power set grows exponentially (2^n)

8. Summary

• Equal sets contain identical elements
• Subsets represent containment (A ⊆ B)
• Power set contains all subsets of a set
• Number of subsets is 2^n