# IGCSE Mathematics Higher: Set Language and Notation

## Scheme of work: IGCSE Foundation: Year 11: Term 1: Set Language and Notation

#### Prerequisite Knowledge

1. Understanding Basic Probability Concepts
• Concept: Knowing how to find the probability of an event occurring.
• Example: $\text{Probability of drawing an Ace from a standard deck of cards: } P(\text{Ace}) = \frac{4}{52} = \frac{1}{13}.$
2. Using Tree Diagrams
• Concept: Understanding how to use tree diagrams to represent and organize the possible outcomes of events.
• Example: $\text{Drawing a tree diagram to show the outcomes of flipping two coins.}$ $\text{First flip: } H, T \quad \text{Second flip: } HH, HT, TH, TT$
3. Using Two-Way Tables
• Concept: Understanding how to use two-way tables to organize data and calculate probabilities.
• Example: $\text{A two-way table showing the number of students who prefer different types of sports:}$ $\begin{array}{|c|c|c|c|} \hline & \text{Football} & \text{Basketball} & \text{Total} \\ \hline \text{Boys} & 20 & 15 & 35 \\ \hline \text{Girls} & 10 & 5 & 15 \\ \hline \text{Total} & 30 & 20 & 50 \\ \hline \end{array}$

#### Success Criteria

1. Understand Sets Defined in Algebraic Terms and Use Subsets
• Objective: Recognize and define sets using algebraic notation and understand the concept of subsets.
• Example: $\text{If } A = \{x \mid x \text{ is an even number}\} \text{, then } B = \{2, 4, 6\} \text{ is a subset of } A \text{ ( } B \subseteq A \text{)}.$
2. Use Venn Diagrams to Represent Sets and the Number of Elements in Sets
• Objective: Draw and interpret Venn diagrams to represent sets and the number of elements in each set.
• Example: $\text{Draw a Venn diagram to represent the sets } A = \{1, 2, 3, 4\} \text{ and } B = \{3, 4, 5, 6\}.$ $\text{Show the intersection } A \cap B = \{3, 4\} \text{ and the union } A \cup B = \{1, 2, 3, 4, 5, 6\}.$
3. Use the Notation $$n(A)$$ for the Number of Elements in the Set $$A$$
• Objective: Understand and use the notation $$n(A)$$ to denote the number of elements in set $$A$$.
• Example: $\text{For set } A = \{2, 4, 6, 8\}, \text{ the number of elements is } n(A) = 4.$
4. Use Sets in Practical Situations
• Objective: Apply the concepts of sets to solve practical problems, such as organizing data or determining probabilities.
• Example: $\text{Use sets to represent the students who play football (F) and basketball (B) in a class:}$ $F = \{1, 2, 3, 4, 5\}, B = \{4, 5, 6, 7\}.$ $\text{Find the students who play both sports } (F \cap B = \{4, 5\}) \text{ and the students who play either sport } (F \cup B = \{1, 2, 3, 4, 5, 6, 7\}).$

#### Key Concepts

1. Understand Sets Defined in Algebraic Terms and Use Subsets
• Concept: A set is a collection of distinct objects, and a subset is a set contained within another set.
• Example: $\text{If } A = \{x \mid x \text{ is an even number}\}, \text{ then } B = \{2, 4, 6\} \text{ is a subset of } A \text{ ( } B \subseteq A \text{)}.$
2. Use Venn Diagrams to Represent Sets and the Number of Elements in Sets
• Concept: Venn diagrams visually represent sets, showing the relationships between different sets, such as intersections and unions.
• Example: $\text{A Venn diagram for sets } A = \{1, 2, 3, 4\} \text{ and } B = \{3, 4, 5, 6\} \text{ shows } A \cap B = \{3, 4\} \text{ and } A \cup B = \{1, 2, 3, 4, 5, 6\}.$
3. Use the Notation $$n(A)$$ for the Number of Elements in the Set $$A$$
• Concept: The notation $$n(A)$$ represents the number of elements in a set $$A$$.
• Example: $\text{For set } A = \{2, 4, 6, 8\}, \text{ the number of elements is } n(A) = 4.$
4. Use Sets in Practical Situations
• Concept: Sets can be used to organize data, solve problems, and calculate probabilities in practical situations.
• Example: $\text{Use sets to represent students who play football (F) and basketball (B) in a class:}$ $F = \{1, 2, 3, 4, 5\}, B = \{4, 5, 6, 7\}.$ $\text{Find the students who play both sports } (F \cap B = \{4, 5\}) \text{ and the students who play either sport } (F \cup B = \{1, 2, 3, 4, 5, 6, 7\}).$

#### Common Misconceptions

1. Understand Sets Defined in Algebraic Terms and Use Subsets
• Common Mistake: Confusing the elements of the set with the set itself or misunderstanding the concept of subsets.
• Example: $\text{Incorrect: Assuming that if } A = \{2, 4, 6\} \text{ and } B = \{2, 4, 6, 8\}, \text{ then } A \supseteq B.$ $\text{Correct: } A \subseteq B.$
2. Use Venn Diagrams to Represent Sets and the Number of Elements in Sets
• Common Mistake: Incorrectly representing the intersection and union of sets in a Venn diagram.
• Example: $\text{Incorrect: Drawing a Venn diagram for sets } A = \{1, 2, 3\} \text{ and } B = \{3, 4, 5\} \text{ with } A \cap B = \{1, 2, 3, 4, 5\}.$ $\text{Correct: } A \cap B = \{3\}.$
3. Use the Notation $$n(A)$$ for the Number of Elements in the Set $$A$$
• Common Mistake: Miscounting the number of elements in a set or including duplicates.
• Example: $\text{Incorrect: For set } A = \{2, 4, 6, 8, 8\}, \text{ the number of elements is } n(A) = 5.$ $\text{Correct: } n(A) = 4 \text{ (duplicates should not be counted).}$
4. Use Sets in Practical Situations
• Common Mistake: Incorrectly interpreting set operations or failing to properly apply sets to solve practical problems.
• Example: $\text{Incorrect: Using sets to represent students who play football (F) and basketball (B) in a class:}$ $F = \{1, 2, 3, 4, 5\}, B = \{4, 5, 6, 7\}.$ $\text{Finding the students who play both sports incorrectly: } F \cap B = \{4, 5, 6, 7\}.$ $\text{Correct: } F \cap B = \{4, 5\}.$

### Mr Mathematics Blog

#### Estimating Solutions by Rounding to a Significant Figure

Explore key concepts, FAQs, and applications of estimating solutions for Key Stage 3, GCSE and IGCSE mathematics.

#### Understanding Equivalent Fractions

Explore key concepts, FAQs, and applications of equivalent fractions in Key Stage 3 mathematics.

#### Transforming Graphs Using Function Notation

Guide for teaching how to transform graphs using function notation for A-Level mathematics.