IB MYP 4-5 Maths- Sets, including notation and operations - Study Notes - New Syllabus
IB MYP 4-5 Maths- Sets, including notation and operations – Study Notes
Standard
- Sets, including notation and operations
IB MYP 4-5 Maths- Sets, including notation and operations – Study Notes – All topics
Sets: Definition and Notation
Sets: Definition and Representation
A set is a well-defined collection of distinct objects, called elements. Sets are represented by capital letters and their elements by curly brackets.
Examples: \( A = \{1, 2, 3\},\ B = \{\text{apple}, \text{banana}, \text{grape}\} \)
Elements and Cardinality
- An element of a set is an individual object in that set.
- Notation: \( x \in A \) means “x is an element of A”.
- Cardinality: The number of elements in a set. Denoted by \( n(A) \).
- Example: For \( A = \{1, 2, 3, 4\} \), \( n(A) = 4 \).
Representing Sets
- Roster/Tabular Form: List elements inside curly braces. Example: \( A = \{2, 4, 6, 8\} \)
- Set-Builder Form: Describe properties of elements. Example: \( A = \{x : x \text{ is an even number } \le 8\} \)
- Venn Diagram: Use circles inside a rectangle (universal set) to represent relationships among sets.
Types of Sets
| Type | Description | Example |
|---|---|---|
| Empty Set (\( \emptyset \)) | No elements | \( \{ \} \) |
| Finite Set | Countable number of elements | \( \{1, 2, 3\} \) |
| Infinite Set | Uncountable elements | \( \{1, 2, 3, \dots\} \) |
| Equal Sets | Same elements | \( \{a,b\} = \{b,a\} \) |
| Subset | Every element of A is in B | \( A \subset B \) |
Common Set Notation:
| Notation | Meaning |
|---|---|
| \( \in \) | Element of |
| \( \notin \) | Not an element of |
| \( \subset \) | Subset of |
| \( \cup \) | Union |
| \( \cap \) | Intersection |
| \( U \) | Universal Set |
| \( \emptyset \) | Empty Set |
Venn Diagram Representation:
- Universal set \( U \) is represented by a rectangle.
- Each set (A, B, C) is shown as a circle inside the rectangle.
- \( A \cup B \): Shaded area covering both circles (Union).
- \( A \cap B \): Overlapping region of two circles (Intersection).
- \( A – B \): Area in A but outside B (Difference).
- \( A’ \): Area outside A but inside the rectangle (Complement).
Operations on Sets:
- Union (\( A \cup B \)): All elements in A or B or both.
- Intersection (\( A \cap B \)): All elements common to A and B.
- Difference (\( A – B \)): Elements in A but not in B.
- Complement (\( A’ \)): All elements in universal set U but not in A.
For Three Sets (A, B, C):
- \( A \cup B \cup C \): All elements in A or B or C.
- \( A \cap B \cap C \): Elements common to all three sets.
- \( (A \cup B) \cap C \): Elements in C and at least one of A or B.
Venn Diagram Representation:
Properties of Sets
- Commutative: \( A \cup B = B \cup A,\ A \cap B = B \cap A \)
- Associative: \( A \cup (B \cup C) = (A \cup B) \cup C \)
- Distributive: \( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)
- Identity: \( A \cup \emptyset = A,\ A \cap U = A \)
- Complement Laws: \( A \cup A’ = U,\ A \cap A’ = \emptyset \)
Example:
If \( A \) and \( B \) are two sets such that \( n(A) = 12 \), \( n(B) = 8 \), and \( n(A \cup B) = 15 \), find \( n(A \cap B) \).
▶️Answer/Explanation
Step 1: Use the formula: \( n(A \cup B) = n(A) + n(B) – n(A \cap B) \)
Step 2: Substitute values: \( 15 = 12 + 8 – n(A \cap B) \)
Step 3: \( n(A \cap B) = 12 + 8 – 15 = 5 \)
Answer: \( n(A \cap B) = 5 \).
Example:
If \( A = \{2, 4, 6, 8\} \), \( B = \{4, 6, 8, 10, 12\} \), and the universal set \( U = \{2, 4, 6, 8, 10, 12, 14\} \), find:
(a) \( A \cup B \),
(b) \( A \cap B \),
(c) \( A’ \) (complement of A).
▶️Answer/Explanation
(a) \( A \cup B = \{2, 4, 6, 8, 10, 12\} \)
(b) \( A \cap B = \{4, 6, 8\} \)
(c) \( A’ = \{10, 12, 14\} \) (elements in U but not in A).
Example:
Express the set \( A = \{3, 6, 9, 12, 15, 18\} \) in set-builder notation.
▶️Answer/Explanation
Observation: These numbers are multiples of 3 up to 18.
So, \( A = \{ x : x = 3n,\ 1 \leq n \leq 6,\ n \in \mathbb{N} \} \).
Example:
Given \( A = \{1, 2, 3, 4\} \), \( B = \{3, 4, 5, 6\} \). Find:
(a) \( A \cup B \)
(b) \( A \cap B \)
(c) \( A – B \)
▶️Answer/Explanation
(a) \( A \cup B = \{1, 2, 3, 4, 5, 6\} \)
(b) \( A \cap B = \{3, 4\} \)
(c) \( A – B = \{1, 2\} \)
Example:
In a survey of 100 students:
• 50 like tea (T)
• 40 like coffee (C)
• 20 like both
(a) Represent this using a Venn diagram.
(b) How many like only tea?
▶️Answer/Explanation
Step 1: Use two-set Venn diagram with overlap = 20.
Tea only: \( 50 – 20 = 30 \).
Coffee only: \( 40 – 20 = 20 \).
Answer: 30 students like only tea.
Example:
In a class:
• 10 study Math (M)
• 8 study Science (S)
• 6 study English (E)
• 3 study all three subjects
• 2 study Math and Science only
• 1 studies Science and English only
How many study only Math?
▶️Answer/Explanation
Step 1: Start with intersection of all three = 3.
Math and Science only: 2 (already includes all three, so add separately).
Total in Math: 10 = only Math + 2 + 3.
So only Math: 10 – (2 + 3) = 5.