Sets
1. Introduction to Sets
A set is a well-defined collection of distinct objects. The objects are called elements or members of the set.
- A set is denoted by capital letters: A, B, C, ...
- Elements are written in curly brackets: { }
- Symbol 'belongs to':
is an element of - Symbol 'does not belong to':
is not an element of
Example: A = {red, blue, yellow}.
red is an element of A. green is not an element of A.
Well-defined means we can clearly say whether an object belongs to the set or not.
Common Mistake: Calling 'a collection of beautiful paintings' a set. 'Beautiful' is subjective, so it is NOT well-defined.
2. Representing Sets
Roster Form (Tabular Form)
List all elements separated by commas, enclosed in curly brackets.
A = {2, 4, 6, 8, 10}
B = {a, e, i, o, u}
Set-Builder Form (Rule Form)
Describe the common property of the elements.
A = {x | x is an even number between 1 and 11}
B = {x | x is a vowel in the English alphabet}
Worked Example: Write {1, 4, 9, 16, 25} in set-builder form.
Each element is a perfect square: 1^2, 2^2, 3^2, 4^2, 5^2.
Set-builder: {x | x = n^2, n is a natural number and n <= 5}.
Exam Focus (2 marks): 'Write the set of letters in the word "MATHEMATICS" in roster form.'
M = {M, A, T, H, E, I, C, S}. Note: repeated letters are listed only once.
3. Types of Sets
Empty (Null) Set
A set with no elements. Denoted by empty set or { }.
Example: A = {x | x is a natural number less than 1}. No natural number is less than 1, so A is empty.
Singleton Set
A set with exactly one element.
Example: B = {5}, C = {x | x is a prime number between 3 and 5}.
Finite Set
A set with a countable number of elements.
Example: D = {2, 4, 6, 8, 10} has 5 elements.
Infinite Set
A set with an unlimited number of elements.
Example: E = {1, 2, 3, 4, ...}, F = {x | x is a point on a line}.
Equal Sets
Two sets are equal if they have exactly the same elements (order does not matter).
A = {1, 2, 3}, B = {3, 1, 2}. A and B are equal.
Equivalent Sets
Two sets have the same number of elements.
A = {a, b, c} and B = {1, 2, 3} are equivalent (both have 3 elements), but not equal.
Common Mistake: Confusing 'equal' with 'equivalent.' Equal sets have identical elements; equivalent sets have the same number of elements.
4. Cardinal Number
The number of elements in a set is its cardinal number, denoted by n(A).
A = {2, 4, 6, 8} => n(A) = 4.
B = { } => n(B) = 0.
5. Venn Diagrams
Venn diagrams use circles (or other shapes) to represent sets visually.
- The universal set is shown as a rectangle.
- Sets are shown as circles inside the rectangle.
Example: Universal set U = {1, 2, 3, 4, 5, 6, 7, 8}, Set A = {2, 4, 6, 8}.
Draw a circle labeled A inside a rectangle labeled U. Place the elements in their correct positions.
6. Comparison Table: Types of Sets
| Type | Definition | Example | n(A) |
|---|---|---|---|
| Empty | No elements | { } | 0 |
| Singleton | Exactly one element | {7} | 1 |
| Finite | Countable elements | {1, 2, 3} | 3 |
| Infinite | Uncountable elements | {1, 2, 3, ...} | Not defined |
| Equal | Same elements | {1,2} and {2,1} | -- |
| Equivalent | Same cardinality | {a,b} and {3,4} | Both 2 |
7. Self-Test
- Which of the following are sets? Justify. (a) Collection of tall students. (b) Collection of prime numbers less than 20.
- Write in roster form: A = {x | x is a month starting with J}.
- Write in set-builder form: B = {2, 3, 5, 7, 11, 13}.
- Identify the type of set: C = {x | x is a natural number and x + 5 = 4}.
- Are these equal or equivalent? A = {p, q, r}, B = {x | x is a letter in 'star'}.
- Find n(D) where D = {factors of 12}.
- Draw a Venn diagram for U = {1,2,3,4,5,6}, A = {2,4,6}.
8. Answers to Self-Test
- (a) Not a set — 'tall' is not well-defined. (b) A set — well-defined: {2,3,5,7,11,13,17,19}.
- {January, June, July}.
- {x | x is a prime number less than 15}.
- No natural number satisfies x + 5 = 4, so C is an empty set.
- B = {s, t, a, r}. Both have 4 elements, so A and B are equivalent. They are NOT equal.
- D = {1, 2, 3, 4, 6, 12}, n(D) = 6.
- Draw rectangle U containing elements 1,2,3,4,5,6. Draw circle A inside containing 2,4,6.
