Set Theory
Sets - Collections of Distinct Objects
A set is a well-defined collection of distinct objects. Those objects are called the elements or members of the set. Set theory is the language that underlies all of modern mathematics – once you understand sets, every other area of maths becomes clearer and more connected.
Set theory was created almost single-handedly by the German mathematician Georg Cantor, who introduced the idea of treating infinite collections as objects in their own right in a series of papers beginning in 1874. His work was so radical for its time – he proved that some infinities are strictly larger than others – that it was fiercely resisted by several contemporaries, yet it went on to become the foundation on which nearly all of modern mathematics is built. By 1908, Ernst Zermelo had begun formalising set theory into a rigorous system of axioms (later extended by Abraham Fraenkel), giving mathematicians a solid logical footing for everything from arithmetic to calculus.
Outside of pure mathematics, the language of sets shows up constantly in everyday computing: a database query that filters rows matching several conditions is really a set-builder description, and search engines combine results using union and intersection logic every time you refine a search.
Notation
Sets are written using curly braces: A = {1, 2, 3, 4}.
The symbol ∈ means “is an element of”: 3 ∈ A.
The symbol ∉ means “is not an element of”: 7 ∉ A.
The cardinality of a set (the number of elements it contains) is written |A|. Here |A| = 4.
Ways to Describe a Set
| Method | Description | Example |
|---|---|---|
| Roster (list) notation | List every element inside curly braces | B = {2, 4, 6, 8, 10} |
| Set-builder notation | Describe the property elements must satisfy | B = {x : x is even, 1 ≤ x ≤ 10} |
| Verbal description | State the rule in plain language | B = the set of even numbers from 2 to 10 |
Special Sets
| Set | Symbol | Description | Example elements |
|---|---|---|---|
| Empty set | ∅ or {} | Contains no elements at all | — |
| Universal set | ξ (or U) | All elements under consideration | All integers, all students in a class |
| Natural numbers | ℕ | Counting numbers | 1, 2, 3, 4, … |
| Integers | ℤ | Whole numbers including negatives and zero | … −2, −1, 0, 1, 2, … |
| Rational numbers | ℚ | Numbers expressible as p/q (q ≠ 0) | 1/2, −3, 0.75 |
| Real numbers | ℝ | All points on the number line | √2, π, −1.5 |
Equal Sets and Equivalent Sets
Equal sets contain exactly the same elements (order and repetition do not matter).
{1, 2, 3} = {3, 1, 2} = {1, 1, 2, 3} (duplicates are ignored).
Equivalent sets have the same number of elements (same cardinality) but not necessarily the same elements.
{1, 2, 3} and {a, b, c} are equivalent because both have cardinality 3.
Worked Examples
A = {M, A, T, H, S}. |A| = 5.
C = {x : x is a multiple of 4, 1 ≤ x ≤ 20}. or C = {4n : n ∈ ℕ, 1 ≤ n ≤ 5}.
(a) True – 5 is listed. (b) True – duplicates are ignored, so both equal {2}. (c) False – |∅| = 0.
Key Takeaways
- A set is a collection of distinct objects called elements.
- Use ∈ for “belongs to” and ∉ for “does not belong to”.
- |A| is the cardinality (number of elements) of set A.
- The empty set ∅ contains no elements; the universal set ξ contains all elements under consideration.
- Order and repetition do not change a set – {1,2,3} = {3,2,1} = {1,1,2,3}.