IGCSE Set Language and Absolute Value: Complete Guide for Cambridge IGCSE Mathematics

IGCSE set language and absolute value are essential topics in Cambridge IGCSE Mathematics that appear in both Paper 2 and Paper 4. Mastering set notation, Venn diagrams, and absolute value equations is crucial for solving problems involving collections of numbers and distance calculations.

This comprehensive IGCSE set language and absolute value guide covers everything you need to know, including set notation symbols, Venn diagram operations, absolute value properties, step-by-step worked examples, common exam questions, and expert tips from Tutopiya’s IGCSE maths tutors. We’ll also show you how to avoid the most common mistakes that cost students valuable marks.

🎯 What you’ll learn: By the end of this guide, you’ll know how to use set notation correctly, solve problems with Venn diagrams, work with absolute values, solve absolute value equations, and structure your working to earn full method marks in IGCSE exams.

Already studying with Tutopiya? Practice these skills with our dedicated IGCSE Algebra practice deck featuring exam-style questions and instant feedback.

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Why IGCSE Set Language and Absolute Value Matter

IGCSE set language and absolute value are important topics that appear regularly in Cambridge IGCSE Mathematics exams. Here’s why they matter:

• High frequency topic: Set notation and absolute value questions appear in most IGCSE maths papers
• Real-world applications: Sets are used in probability, statistics, and data analysis; absolute value represents distance and magnitude
• Exam weight: Typically worth 4-8 marks per paper
• Foundation for advanced topics: Essential for understanding probability, statistics, and inequalities
• Problem-solving skills: Develops logical thinking and systematic problem-solving approaches

Key insight from examiners: Students often lose marks due to incorrect set notation or forgetting that absolute value equations have two solutions. This guide will help you avoid these pitfalls.

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Understanding Sets: The Basics

A set is a collection of distinct objects (called elements or members). Sets are usually denoted by capital letters.

Set Notation

Common symbols:

• ∈ means “is an element of” or “belongs to”
• ∉ means “is not an element of”
• { } curly brackets enclose the elements of a set
• ∅ or {} represents the empty set (a set with no elements)
• n(A) means “the number of elements in set A”
• A ⊆ B means “A is a subset of B” (all elements of A are in B)
• A ⊂ B means “A is a proper subset of B” (A is a subset but not equal to B)
• U represents the universal set (all possible elements)

Examples:

• A = {1, 2, 3, 4, 5} - A is the set containing numbers 1, 2, 3, 4, and 5
• 3 ∈ A - 3 is an element of set A
• 6 ∉ A - 6 is not an element of set A
• n(A) = 5 - Set A has 5 elements

Describing Sets

Three main ways to describe a set:

1. Listing (Roster method): A = {2, 4, 6, 8, 10}
2. Set-builder notation: A = {x : x is an even number between 1 and 11}
   • Read as: “A is the set of all x such that x is an even number between 1 and 11”
3. Description: “A is the set of even numbers from 2 to 10”

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Set Operations

Union (∪)

Definition: The union of sets A and B, written A ∪ B, contains all elements that are in A, in B, or in both.

Example:

• A = {1, 2, 3, 4}
• B = {3, 4, 5, 6}
• A ∪ B = {1, 2, 3, 4, 5, 6}

Intersection (∩)

Definition: The intersection of sets A and B, written A ∩ B, contains only elements that are in both A and B.

Example:

• A = {1, 2, 3, 4}
• B = {3, 4, 5, 6}
• A ∩ B = {3, 4}

Complement (A’ or Aᶜ)

Definition: The complement of set A, written A' or Aᶜ, contains all elements in the universal set that are not in A.

Example:

• U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
• A = {2, 4, 6, 8}
• A' = {1, 3, 5, 7, 9, 10}

Difference (A - B or A \ B)

Definition: The difference of sets A and B, written A - B or A \ B, contains elements that are in A but not in B.

Example:

• A = {1, 2, 3, 4}
• B = {3, 4, 5, 6}
• A - B = {1, 2}

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Venn Diagrams

Venn diagrams are visual representations of sets using overlapping circles. They’re extremely useful for solving set problems.

Basic Venn Diagram Structure

        U (Universal Set)
    ┌─────────────────────┐
    │                     │
    │    ┌──────┐         │
    │    │  A   │         │
    │    │      │         │
    │    │  A∩B │         │
    │    └──┬───┘         │
    │       │             │
    │    ┌──┴───┐         │
    │    │  B   │         │
    │    └──────┘         │
    │                     │
    └─────────────────────┘

Key Regions in a Venn Diagram

• A only: Elements in A but not in B
• B only: Elements in B but not in A
• A ∩ B: Elements in both A and B (intersection)
• A ∪ B: All elements in A or B or both (union)
• A’ or Aᶜ: Everything outside A (complement)
• (A ∪ B)’: Everything outside both A and B

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Step-by-Step Method for Set Problems

1. Identify the sets - What are sets A, B, C, etc.?
2. Draw a Venn diagram - Visualize the problem
3. Fill in known information - Start with intersections, then work outward
4. Use formulas - n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
5. Answer the question - Make sure you answer what’s being asked

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Worked Examples: Sets

Example 1: Basic Set Operations

Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 4, 6, 8}, B = {3, 6, 9}

Find: a) A ∪ B b) A ∩ B c) A' d) A - B

Solution: a) A ∪ B = {2, 3, 4, 6, 8, 9} (all elements in A or B) b) A ∩ B = {6} (elements in both A and B) c) A' = {1, 3, 5, 7, 9, 10} (elements not in A) d) A - B = {2, 4, 8} (elements in A but not in B)

Example 2: Venn Diagram Problem

In a class of 30 students:

• 18 students study Mathematics
• 15 students study Physics
• 8 students study both Mathematics and Physics

Find: a) How many students study only Mathematics? b) How many students study only Physics? c) How many students study at least one of these subjects? d) How many students study neither subject?

Solution: Let M = students studying Mathematics, P = students studying Physics

Given: n(M) = 18, n(P) = 15, n(M ∩ P) = 8, n(U) = 30

a) Only Mathematics: n(M) - n(M ∩ P) = 18 - 8 = 10 b) Only Physics: n(P) - n(M ∩ P) = 15 - 8 = 7 c) At least one: n(M ∪ P) = n(M) + n(P) - n(M ∩ P) = 18 + 15 - 8 = 25 d) Neither: n(U) - n(M ∪ P) = 30 - 25 = 5

Answers: a) 10 students b) 7 students c) 25 students d) 5 students

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Understanding Absolute Value: The Basics

The absolute value of a number is its distance from zero on the number line. It’s always positive or zero.

Notation: |x| means “the absolute value of x”

Definition:

|x| = {
  x,  if x ≥ 0
  -x, if x < 0
}

Examples:

• |5| = 5 (distance from 0 to 5 is 5)
• |-5| = 5 (distance from 0 to -5 is 5)
• |0| = 0

Key property: |x| ≥ 0 for all real numbers x

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Properties of Absolute Value

Property 1: Non-Negativity

|x| ≥ 0 for all x, and |x| = 0 only when x = 0

Property 2: Symmetry

|-x| = |x| (absolute value is symmetric about zero)

Property 3: Multiplication

|xy| = |x| × |y|

Property 4: Division

|x/y| = |x|/|y| (where y ≠ 0)

Property 5: Triangle Inequality

|x + y| ≤ |x| + |y|

Property 6: Square Root

|x| = √(x²)

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Solving Absolute Value Equations

Key principle: If |x| = a (where a ≥ 0), then x = a or x = -a

Method for Solving |expression| = number

1. Set up two equations:
   • expression = number
   • expression = -number
2. Solve both equations
3. Check your solutions - Substitute back into the original equation

Method for Solving |expression| = |other expression|

1. Set up two equations:
   • expression = other expression
   • expression = -(other expression)
2. Solve both equations
3. Check your solutions

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Worked Examples: Absolute Value

Example 1: Simple Absolute Value Equation

Solve: |x - 3| = 7

Solution:

1. Set up two equations:
   • x - 3 = 7 → x = 10
   • x - 3 = -7 → x = -4
2. Check:
   • |10 - 3| = |7| = 7 ✓
   • |-4 - 3| = |-7| = 7 ✓

Answer: x = 10 or x = -4

Example 2: Absolute Value with Coefficients

Solve: |2x + 5| = 13

Solution:

1. Set up two equations:
   • 2x + 5 = 13 → 2x = 8 → x = 4
   • 2x + 5 = -13 → 2x = -18 → x = -9
2. Check:
   • |2(4) + 5| = |13| = 13 ✓
   • |2(-9) + 5| = |-13| = 13 ✓

Answer: x = 4 or x = -9

Example 3: Two Absolute Values

Solve: |x - 2| = |x + 4|

Solution:

1. Set up two equations:
   • x - 2 = x + 4 → -2 = 4 (no solution)
   • x - 2 = -(x + 4) → x - 2 = -x - 4 → 2x = -2 → x = -1
2. Check: |-1 - 2| = |-3| = 3 and |-1 + 4| = |3| = 3 ✓

Answer: x = -1

Example 4: Absolute Value Inequality

Solve: |x - 5| < 3

Solution: For |x - a| < b (where b > 0), the solution is: a - b < x < a + b

Therefore: 5 - 3 < x < 5 + 3, so 2 < x < 8

Answer: 2 < x < 8

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Common Examiner Traps (and How to Dodge Them)

• Forgetting two solutions - Absolute value equations usually have two solutions
• Incorrect set notation - Use proper symbols: ∈, ∪, ∩, '
• Venn diagram errors - Always fill in intersections first, then work outward
• Absolute value with negative - Remember: |x| = -x when x < 0, but |x| itself is never negative
• Not checking solutions - Always verify your answers by substituting back
• Confusing union and intersection - Union (∪) means “or”, intersection (∩) means “and”

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IGCSE Set Language and Absolute Value Practice Questions

Question 1: Set Operations

Given: U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {2, 4, 6, 8}, B = {1, 3, 5, 7}, C = {2, 3, 5, 7}

Find: a) (A ∪ B) ∩ C b) A' ∩ B

Solution: a) A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8} = U (A ∪ B) ∩ C = U ∩ C = C = {2, 3, 5, 7} b) A' = {1, 3, 5, 7} = B A' ∩ B = B ∩ B = B = {1, 3, 5, 7}

Answers: a) {2, 3, 5, 7} b) {1, 3, 5, 7}

Question 2: Venn Diagram Problem

In a survey of 100 students:

• 60 students like football
• 50 students like basketball
• 30 students like both sports

How many students like neither sport?

Solution: Let F = students who like football, B = students who like basketball

n(F ∪ B) = n(F) + n(B) - n(F ∩ B) = 60 + 50 - 30 = 80

Students who like neither: 100 - 80 = 20

Answer: 20 students

Question 3: Absolute Value Equation

Solve: |3x - 7| = 11

Solution:

1. 3x - 7 = 11 → 3x = 18 → x = 6
2. 3x - 7 = -11 → 3x = -4 → x = -4/3

Answer: x = 6 or x = -4/3

Question 4: Absolute Value Inequality

Solve: |2x + 1| ≥ 5

Solution: For |x| ≥ a, the solution is: x ≤ -a or x ≥ a

Therefore: 2x + 1 ≤ -5 or 2x + 1 ≥ 5

• 2x + 1 ≤ -5 → 2x ≤ -6 → x ≤ -3
• 2x + 1 ≥ 5 → 2x ≥ 4 → x ≥ 2

Answer: x ≤ -3 or x ≥ 2

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Frequently Asked Questions About IGCSE Set Language and Absolute Value

What is a set?

A set is a collection of distinct objects (called elements). Sets are usually written with curly brackets: {1, 2, 3, 4, 5}.

What’s the difference between union and intersection?

• Union (∪): Contains all elements that are in either set (or both)
• Intersection (∩): Contains only elements that are in both sets

Example: If A = {1, 2, 3} and B = {3, 4, 5}:

• A ∪ B = {1, 2, 3, 4, 5} (all elements)
• A ∩ B = {3} (only common element)

What is absolute value?

The absolute value of a number is its distance from zero on the number line. It’s always positive or zero. For example, |5| = 5 and |-5| = 5.

Why do absolute value equations have two solutions?

Because |x| = a means “the distance from x to 0 is a”. There are two numbers that are distance a from 0: a and -a.

How do I solve |x - 3| = 7?

Set up two equations:

1. x - 3 = 7 → x = 10
2. x - 3 = -7 → x = -4

So x = 10 or x = -4.

What does n(A) mean?

n(A) means “the number of elements in set A”. For example, if A = {2, 4, 6, 8}, then n(A) = 4.

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Related IGCSE Maths Resources

Strengthen your IGCSE Mathematics preparation with these comprehensive guides:

• IGCSE Exponents and Surds: Complete Guide - Master index laws and surd operations
• IGCSE Trigonometry: Complete SOHCAHTOA Guide - Master right-angled triangle trigonometry
• IGCSE Maths Revision Notes, Syllabus and Preparation Tips - Complete syllabus overview, topic breakdown, and revision strategies
• IGCSE Past Papers Guide - Access free IGCSE past papers and exam resources

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