CIE IGCSE Mathematics (0580) Sets Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Sets Study Notes
Key Concepts:
Core
Understand and use set language, notation and Venn diagrams to describe sets.
Venn diagrams are limited to two sets.
The following set notation will be used:
• n(A) Number of elements in set A
• A′ Complement of set A
• Universal set
• A ∪ B Union of A and B
• A ∩ B Intersection of A and B
Example definition of sets:
A = {x: x is a natural number}
B = {a, b, c, …}
C = {x: a ⩽ x ⩽ b}
Supplement
Understand and use set language, notation and Venn diagrams to describe sets and represent relationships between sets.
Venn diagrams are limited to two or three sets.
The following set notation will be used:
• n(A) Number of elements in set A
• ∈ “is an element of”
• ∉ “is not an element of”
• A′ Complement of set A
• ∅ The empty set
• Universal set
• A ⊆ B A is a subset of B
• A ⊈ B A is not a subset of B
• A ∪ B Union of A and B
• A ∩ B Intersection of A and B
Example definition of sets:
A = {x: x is a natural number}
B = {(x, y): y = mx + c}
C = {x: a ⩽ x ⩽ b}
D = {a, b, c, …}
Set Language, Notation, and Venn Diagrams
Set Language, Notation, and Venn Diagrams
Definition of a Set
A set is a collection of distinct objects or elements. These elements can be numbers, letters, or any well-defined items.
Sets are usually denoted by capital letters like \( A, B, C \), and elements are listed within curly brackets.
Example: \( A = \{2, 4, 6, 8\} \)
Set Notation and Symbols
| Symbol | Meaning |
|---|---|
| \( n(A) \) | Number of elements in set \( A \) |
| \( A’ \) or \( A^c \) | Complement of set \( A \) (elements not in \( A \)) |
| \( A \cup B \) | Union — all elements in \( A \) or \( B \) or both |
| \( A \cap B \) | Intersection — elements common to both \( A \) and \( B \) |
| \( A \setminus B \) | Difference — elements in \( A \) but not in \( B \) |
| \( \xi \) | Universal set — the set containing all elements under consideration |
| \( \subseteq \) | Subset — all elements of one set are contained in another |
| \( \emptyset \) | Empty set — a set with no elements |
| \( \in \) | “is an element of” (e.g., \( 3 \in A \)) |
| \( \notin \) | “is not an element of” (e.g., \( 7 \notin A \)) |
Describing Sets
Sets can be described in two main ways:
- List form: \( B = \{1, 3, 5, 7\} \)
- Set-builder form: \( B = \{x : x \text{ is an odd number less than } 10\} \)
Special Sets
These commonly used sets have specific symbols:
- \( \mathbb{N} \): Natural numbers
- \( \mathbb{Z} \): Integers
- \( \mathbb{Q} \): Rational numbers
- \( \mathbb{R} \): Real numbers
- \( \mathbb{P} \): Prime numbers
Venn Diagrams
Venn diagrams are visual representations of sets using overlapping circles. Each set is shown as a circle, and the universal set \( \xi \) is usually shown as a rectangle containing all relevant elements.
| Image | Set Notation | Description |
|---|---|---|
 | \( A \) | Only elements in set A |
 | \( A’ \) or \( A^c \) | Complement of A (everything not in A) |
 | \( A \cap B = \emptyset \) | Disjoint sets (no common elements) |
 | \( B \subset A \) | B is a proper subset of A |
 | \( A \cap B \) | Elements common to both A and B |
 | \( A \cup B \) | Union of A and B (all elements in A or B or both) |
Set Relationships and Vocabulary
- Equal Sets: Two sets are equal if they have exactly the same elements.
- Subsets: \( A \subseteq B \) means every element of \( A \) is also in \( B \)
- Disjoint Sets: Sets with no elements in common (i.e., \( A \cap B = \emptyset \))
Example:
Let the universal set be \( \xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \), and \( A = \{2, 4, 6, 8\} \).
List the elements of \( A’ \), the complement of set \( A \).
▶️ Answer/Explanation
Step 1: Identify all elements in the universal set \( \xi \).
\( \xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)
Step 2: Identify elements in set \( A \).
\( A = \{2, 4, 6, 8\} \)
Step 3: The complement \( A’ \) is everything in \( \xi \) that is not in \( A \).
\( A’ = \{1, 3, 5, 7, 9, 10\} \)
Example:
Let \( A = \{2, 4, 6\} \), \( B = \{4, 5, 6, 7\} \).
Find \( A \cup B \) and \( A \cap B \).
▶️ Answer/Explanation
Union: Combine all elements from both sets (no repetition).
\( A \cup B = \{2, 4, 5, 6, 7\} \)
Intersection: Elements common to both sets.
\( A \cap B = \{4, 6\} \)
Example:
In a class of 40 students:
- 18 study Mathematics
- 20 study Physics
- 7 study both Mathematics and Physics
How many study neither subject?
▶️ Answer/Explanation
Step 1: Use a Venn diagram (or formula):
\( n(A \cup B) = n(A) + n(B) – n(A \cap B) \) \( = 18 + 20 – 7 = 31 \)
Step 2: Total students = 40
So, number who study neither = \( 40 – 31 = 9 \)
Example:
In a Venn diagram with two sets \( A \) and \( B \), describe and shade the region representing \( A \cup B’ \).
▶️ Answer/Explanation
Step 1: Understand the expression.
\( B’ \) is everything outside set \( B \).
\( A \cup B’ \) includes everything in set \( A \), or outside set \( B \), or both.
Step 2: Region includes:
- All of set \( A \)
- Everything outside \( B \) (including parts not in \( A \))
- Only the overlapping part that lies in both \( A \) and outside \( B \)
This region includes everything except the part that is only in \( B \) and not in \( A \).