Set theory is a cornerstone of modern mathematics, playing a crucial role in algebra, probability, computer science, and mathematical logic. This article explores fundamental set operations, key properties such as associativity and closure, De Morgan's laws, Cartesian product, and types of numerical sets.
1. Set Operations
Union (∪)
The union of two sets includes all elements from both sets:
A = {1, 2, 3}, B = {3, 4, 5} A ∪ B = {1, 2, 3, 4, 5}
Properties:
Associativity:
A ∪ (B ∪ C) = (A ∪ B) ∪ C
Commutativity:
A ∪ B = B ∪ A
Identity Element: Union with the empty set does not change a set:
A ∪ ∅ = A
Intersection (∩)
Intersection includes only the common elements of two sets:
A ∩ B = {3}
Properties:
Associativity:
A ∩ (B ∩ C) = (A ∩ B) ∩ C
Commutativity:
A ∩ B = B ∩ A
Identity Element:
A ∩ U = A (if U is the universal set) A ∩ ∅ = ∅
Difference (−)
The difference operation removes elements of one set from another:
A − B = {1, 2} B − A = {4, 5}
Properties:
A − A = ∅ (removes all elements)
A − ∅ = A (set remains unchanged)
2. De Morgan's Laws
De Morgan's laws express relationships between union, intersection, and set complement:
(A ∪ B)ᶜ = Aᶜ ∩ Bᶜ (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ
These laws demonstrate that the complement of a union is the intersection of complements and vice versa.
3. Cartesian Product (×)
The Cartesian product creates ordered pairs of all elements from two sets:
A = {x, y}, B = {1, 2} A × B = {(x,1), (x,2), (y,1), (y,2)}
Properties:
Number of elements:
|A × B| = |A| × |B|
Non-commutativity:
A × B ≠ B × A
4. Set Equality and Equivalence
Set Equality
Two sets are equal if they contain exactly the same elements:
A = {1, 2, 3}, B = {3, 2, 1} A = B (order does not matter)
Set Equivalence
Two sets are equivalent if they have the same number of elements:
A = {x, y, z}, B = {1, 2, 3} |A| = |B| → A ≈ B
5. Set Closure Property
A set is closed under an operation if applying the operation to its elements results in another element of the same set.
Examples:
Integers are closed under addition:
∀ a, b ∈ ℤ → (a + b ∈ ℤ)
Natural numbers are not closed under subtraction:
3 - 5 ∉ ℕ
6. Types of Numerical Sets
Categories of Numerical Sets and Their Properties
| Set Name | Symbol | Properties |
| Natural Numbers | ℕ | {0, 1, 2, 3, ...} - Closed under addition and multiplication |
| Integers | ℤ | {..., -2, -1, 0, 1, 2, ...} - Closed under addition, subtraction, and multiplication |
| Rational Numbers | ℚ | Includes all fractions like 1/2 - Closed under four basic operations |
| Irrational Numbers | ℚᶜ | Includes non-repeating decimals like √2 and π |
| Real Numbers | ℝ | Includes all numbers: natural, integers, rational, and irrational |
