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.
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}.
Practice Questions
- List the elements of D = {x : x is a prime number less than 20}.
- Write in set-builder notation: E = {1, 4, 9, 16, 25, 36}.
- State the cardinality of each: (a) F = {vowels in the alphabet}; (b) G = ∅; (c) H = {x : x is a whole number, 1 ≤ x ≤ 100}.
- True or false: (a) {a, b, c} = {c, b, a}; (b) 0 ∈ ℕ (natural numbers); (c) {0} = ∅.
- Give an example of two sets that are equivalent but not equal.