Mathematical Sets: Definitions, Formulas, and Applications

1. Set Operations

Union (∪)

The union of two sets includes all elements from both:

A = {1, 2, 3}, B = {3, 4, 5}
A ∪ B = {1, 2, 3, 4, 5}

• Associativity: A ∪ (B ∪ C) = (A ∪ B) ∪ C
• Commutativity: A ∪ B = B ∪ A
• Identity: A ∪ ∅ = A

Intersection (∩)

Includes only elements common to both sets:

A ∩ B = {3}

• Associativity: A ∩ (B ∩ C) = (A ∩ B) ∩ C
• Commutativity: A ∩ B = B ∩ A
• Identity: A ∩ U = A; A ∩ ∅ = ∅

Difference (−)

Removes elements of one set from another:

A − B = {1, 2}
B − A = {4, 5}

• A − A = ∅
• A − ∅ = A

2. De Morgan’s Laws

Show how complements interact with union and intersection:

(A ∪ B)ᶜ = Aᶜ ∩ Bᶜ
(A ∩ B)ᶜ = Aᶜ ∪ Bᶜ

3. Cartesian Product (×)

Creates ordered pairs from elements in two sets:

A = {x, y}, B = {1, 2}
A × B = {(x,1), (x,2), (y,1), (y,2)}

• Size: |A × B| = |A| × |B|
• Not commutative: A × B ≠ B × A

4. Set Equality and Equivalence

Equality

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

Order does not affect equality.

Equivalence

A = {x, y, z}, B = {1, 2, 3}
|A| = |B| → A ≈ B

Same number of elements implies equivalence.

5. Closure Property

A set is closed under an operation if applying the operation yields a result in the same set:

• ℤ is closed under addition: ∀ a, b ∈ ℤ → a + b ∈ ℤ
• ℕ is not closed under subtraction: 3 − 5 ∉ ℕ

6. Types of Numerical Sets

Set	Symbol	Properties
Natural Numbers	ℕ	{0, 1, 2, 3, ...}; closed under + and ×
Integers	ℤ	{..., −2, −1, 0, 1, 2, ...}; closed under +, −, ×
Rational Numbers	ℚ	Includes fractions; closed under all four operations
Irrational Numbers	ℚᶜ	Non-repeating decimals like √2, π
Real Numbers	ℝ	Includes all above: ℕ, ℤ, ℚ, ℚᶜ

Conclusion

Set theory underpins many mathematical and computational concepts. By understanding fundamental operations, properties like closure and equality, and the structure of numerical sets, learners gain insight into logic, algebra, and data modeling that applies across disciplines.