International Mathematics (0607) | Sets

Welcome to the World of Sets! (IGCSE 0607 - Number Section)

Hello mathematicians! This chapter introduces you to the concept of Sets. Don't worry, this isn't complicated math—it's just a precise way of grouping things. Think of sets as super-organized lists or collections.

Why is this important? Sets provide the fundamental language for advanced mathematics, logic, and even computer science. Mastering the notation will help you define groups of numbers and solve probability problems (which you'll see later in the course!) much more clearly. Let's get organized!

1. Defining Sets and Basic Notation

A set is simply a well-defined collection of distinct objects. These objects are called elements or members of the set.

1.1 Ways to Describe a Set

Method 1: Listing the Elements (Roster Method)

You list all the elements separated by commas, contained within curly braces \(\{\}\).
Example: The set A containing the first four prime numbers is:
\(A = \{2, 3, 5, 7\}\)

Method 2: Set Builder Notation (Rule Method)

This describes the elements using a rule. This is very common when dealing with large or infinite sets.
Example: The set B of all natural numbers greater than 10.
\(B = \{x \mid x \text{ is a natural number and } x > 10\}\)
(Read this as: "B is the set of all elements \(x\) such that \(x\) is a natural number and \(x\) is greater than 10.")

The syllabus often uses this notation for defining number ranges:
Example: \(C = \{x \mid 2 < x < 5\}\). If \(x\) must be an integer, then \(C = \{3, 4\}\).

1.2 Counting Elements: Cardinality \(n(A)\)

The cardinal number of a set A, written as \(n(A)\), is the number of elements in that set.

• If \(A = \{ \text{red, blue, green} \}\), then \(n(A) = 3\).
• If \(B = \{10, 20, 30, 40, 50\}\), then \(n(B) = 5\).

2. Key Set Notation and Relationships (Core and Extended)

2.1 The Universal Set \(\mathcal{U}\)

The Universal Set, denoted by \(\mathcal{U}\) (a fancy 'U'), is the set that contains all the elements relevant to a particular problem. Think of it as the boundary or the "entire universe" for the calculation you are currently doing.

• Analogy: If you are studying students in your IGCSE Maths class, \(\mathcal{U}\) is "all students in the IGCSE Maths class."

2.2 Membership Notation (Extended Only: E1.2)

(Extended students need to know this notation.)
We use special symbols to show if an element belongs to a set:

• \(\in\): "is an element of" or "belongs to."
  Example: If \(A = \{1, 2, 3\}\), then \(2 \in A\).
• \(\notin\): "is not an element of."
  Example: If \(A = \{1, 2, 3\}\), then \(5 \notin A\).

2.3 Subsets and Non-Subsets (Extended Only: E1.2)

(Extended students need to know this notation.)
Set B is a subset of Set A, written as \(B \subset A\), if every single element of B is also an element of A.

• Analogy: If Set A is "All cats" and Set B is "All black cats," then \(B \subset A\).
• \(\subset\): "is a subset of."
  Example: If \(X = \{1, 2, 3, 4\}\) and \(Y = \{1, 4\}\), then \(Y \subset X\).
• \(\not\subset\): "is not a subset of."
  Example: If \(Z = \{5, 6\}\), then \(Z \not\subset X\) (because 5 and 6 are not in X).

2.4 The Empty Set \(\emptyset\) (Extended Only: E1.2)

(Extended students need to know this notation.)
The empty set, denoted by \(\emptyset\) or \(\{\}\), is a set that contains absolutely no elements.
Example: The set of all IGCSE students who are 2 years old is \(\emptyset\).

3. Set Operations and Venn Diagrams

Set operations show how sets can interact with each other. We use diagrams called Venn Diagrams to visualize these operations.

3.1 The Complement of a Set \(A'\)

The complement of A, written as \(A'\), is the set of all elements in the Universal Set (\(\mathcal{U}\)) that are not in A.

• Analogy: If \(\mathcal{U}\) is "All students in school," and A is "Students wearing trainers," then \(A'\) is "Students not wearing trainers."
• In a Venn diagram, \(A'\) is the area outside the circle A but still inside the universal box \(\mathcal{U}\).

3.2 The Intersection of Two Sets \(A \cap B\)

The intersection of two sets A and B, written as \(A \cap B\), is the set of elements that are in both A AND B.

• Memory Aid: Think of the symbol \(\cap\) looking like a bridge where two roads meet. This is the overlap!
• Example: If \(A = \{1, 2, 3, 4\}\) and \(B = \{3, 4, 5, 6\}\), then \(A \cap B = \{3, 4\}\).
• In a Venn diagram, \(A \cap B\) is the overlapping area between the two circles.

3.3 The Union of Two Sets \(A \cup B\)

The union of two sets A and B, written as \(A \cup B\), is the set of elements that are in A OR B (or both).

• Memory Aid: Think of the symbol \(\cup\) as a cup used to "Unite" or collect everything.
• Example: If \(A = \{1, 2, 3, 4\}\) and \(B = \{3, 4, 5, 6\}\), then \(A \cup B = \{1, 2, 3, 4, 5, 6\}\). (Note: We do not list elements twice!)
• In a Venn diagram, \(A \cup B\) is the entire area covered by both circles A and B.

Did you know? If \(A \cap B = \emptyset\), it means the sets share no common elements. We call these sets disjoint sets. They look like two separate circles in a Venn diagram.

3.4 Venn Diagrams (Visualization)

Two Sets (Core and Extended)

The syllabus requires you to understand and use Venn diagrams involving up to two sets (for Core) or up to three sets (for Extended).

A two-set Venn diagram divides the universal set into four distinct regions:

1. Elements in A only: \(A \cap B'\)
2. Elements in B only: \(A' \cap B\)
3. Elements in both A and B (Intersection): \(A \cap B\)
4. Elements in neither A nor B (Complement of the Union): \((A \cup B)'\)

When solving problems using Venn diagrams, always start by placing the number of elements in the intersection first!

Three Sets (Extended Only: E1.2)

(Extended students must handle three overlapping sets: A, B, and C.)
When dealing with three sets, the Venn diagram has three overlapping circles, creating many more regions (8 regions total).

You need to be able to identify specific regions, such as:

• The central region: \(A \cap B \cap C\) (Elements in A AND B AND C).
• Elements in A and B, but not C: \(A \cap B \cap C'\).
• Elements in A only: \(A \cap B' \cap C'\).

Just like with two sets, the trick to solving three-set problems is to start from the most central overlap (\(A \cap B \cap C\)) and work your way outwards.

Key Takeaways and Summary

Sets might seem abstract, but they are just groups defined by rules!