Mathematics (Specification B) | Sets

Welcome to the World of Sets!

Hello future mathematicians! Sets are a fundamental building block in almost every area of math. Don't worry if this chapter seems abstract at first; sets are really just a clever, standardized way of organizing and grouping things. Think of a set as a perfectly curated collection, like a playlist of your favourite songs, or the specific ingredients needed for a recipe.

In this chapter, we will learn the special language and rules used to define, combine, and compare these collections. Mastering this notation will help you tackle complex problems in logic, probability, and beyond!

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1. Defining Sets and Basic Notation

What is a Set?

A Set is a collection of distinct objects. The objects within the set are called elements or members.

Example: The set of primary colours.
The elements are {Red, Yellow, Blue}.

Methods of Defining a Set

A. Listing Elements (Roster Method)

We list the elements and separate them with commas, enclosed in curly brackets \(\{\}\).

• If Set A is the set of even numbers less than 10:
  \(A = \{2, 4, 6, 8\}\)

Important Notation:

• Belongs to (\(\in\)): Used to show an element is part of a set.
  Example: \(4 \in A\) (4 belongs to A)
• Does not belong to (\(\notin\)): Used to show an element is NOT part of a set.
  Example: \(5 \notin A\) (5 does not belong to A)

B. Set Builder Notation (Description Method)

This method describes the rule that all elements must follow. It is essential when dealing with sets that are too large to list.

The structure looks like this: \(\{x : x \text{ satisfies a certain property}\}\)

The colon (\(:\)) or sometimes a vertical bar (\(| \)) means "such that".

• If B is the set of all integers greater than 5:
  \(B = \{x : x \in \mathbb{Z}, x > 5\}\)
  (Read as: "B is the set of elements x, such that x is an integer, and x is greater than 5.")

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2. Special Types of Sets

Universal Set (\(\mathcal{E}\))

The Universal Set (\(\mathcal{E}\)) is the set that contains all the elements relevant to a particular context or problem. Think of it as the boundary of your current mathematical world.

Analogy: If you are talking about the students in your school, the Universal Set is all the students in the school.

Empty Set (or Null Set)

The Empty Set is a set containing NO elements. It is denoted by \(\emptyset\) or \(\{\}\).

Example: If \(C\) is the set of students in your class who are 150 years old, \(C = \emptyset\).

Finite and Infinite Sets

• A Finite Set is a set whose elements can be counted, and the counting ends. (e.g., The letters in the word "MATH").
• An Infinite Set is a set whose elements cannot be counted, or the counting never ends. (e.g., The set of all whole numbers \(\mathbb{W}\) or the set of all points on a line).

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3. Relationships Between Sets (Subsets)

We often need to compare sets to see if one is part of another.

Subset (\(\subseteq\))

Set A is a Subset of Set B if every element in A is also an element in B.

Notation: \(A \subseteq B\)
Analogy: Your history class (A) is a subset of all the students in your school (B).

Crucial Rule: The empty set \(\emptyset\) is a subset of every set. Every set is also a subset of itself.

Proper Subset (\(\subset\))

Set A is a Proper Subset of Set B if A is a subset of B, AND B contains at least one element that is NOT in A (meaning \(A \neq B\)).

Notation: \(A \subset B\)
Think of it: A is strictly smaller than B.

Let's look at an example:
\(B = \{1, 2, 3, 4, 5\}\)
\(A = \{2, 3\}\)
\(C = \{1, 2, 3, 4, 5\}\)

\(A \subset B\) (A is a proper subset of B)
\(C \subseteq B\) (C is a subset of B, and in this case, C equals B)

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4. Set Operations

Set operations allow us to combine or subtract sets based on common elements.

Intersection (\(\cap\))

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.

Mnemonic: The symbol \(\cap\) looks like a bridge or an arch—it connects the common area.

Example:
\(A = \{1, 2, 3, 4\}\)
\(B = \{3, 4, 5, 6\}\)
\(A \cap B = \{3, 4\}\)

If two sets have no elements in common, their intersection is the empty set. They are called Disjoint Sets (e.g., \(A \cap B = \emptyset\)).

Union (\(\cup\))

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. We list every unique element present in either set.

Mnemonic: The symbol \(\cup\) looks like a cup—it holds everything!

Example (using sets A and B above):
\(A \cup B = \{1, 2, 3, 4, 5, 6\}\)

Complement (\(A'\))

The Complement of set A, written as \(A'\) or \(A^c\), is the set of all elements in the Universal Set (\(\mathcal{E}\)) that are NOT in A.

Example:
\(\mathcal{E} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\)
\(A = \{1, 3, 5, 7, 9\}\) (Odd numbers)
\(A' = \{2, 4, 6, 8, 10\}\) (Even numbers)

Set Difference (\(A \setminus B\) or \(A - B\))

The Set Difference is the set of elements that are in A but are not in B.

\(A \setminus B = A \cap B'\)
Example (using sets A and B above):
\(A = \{1, 2, 3, 4\}\)
\(B = \{3, 4, 5, 6\}\)
\(A \setminus B = \{1, 2\}\) (We take A and remove any parts that B shares.)

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5. Visualizing Sets with Venn Diagrams

Venn Diagrams are graphical tools that use overlapping circles to show relationships between sets.

• The rectangle represents the Universal Set (\(\mathcal{E}\)).
• The circles represent the individual sets (A, B, C).

Step-by-Step Shading (Two Sets)

Shading the region corresponding to the notation helps cement your understanding.

1. \(A \cap B\): Shade the area where the two circles overlap (the intersection). This is the region in A AND in B.
2. \(A \cup B\): Shade everything in A and everything in B. This is the region in A OR in B.
3. \(A'\): Shade everything outside of circle A (but still inside the Universal rectangle).
4. \((A \cup B)'\): Shade everything outside both circles.
5. \(A \cap B'\): Shade the part of circle A that does NOT overlap with B. (This is the Set Difference \(A \setminus B\)).

Venn Diagrams with Three Sets (A, B, C)

When working with three sets, the Venn diagram has three overlapping circles, creating 8 distinct regions. Problems in Specification B often require you to identify or fill in these specific regions.

• The very center region: \(A \cap B \cap C\) (Elements common to all three).
• The region shared only by A and B (excluding C): \(A \cap B \cap C'\)
• The region only in A (excluding B and C): \(A \cap B' \cap C'\)

Tip for shading complex regions: Use light pencil shading or different colours for each component of the expression (e.g., shade A' in blue, shade B in red), and the required region will be the double-shaded area.

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6. Cardinality and Problem Solving

Cardinality (\(n(A)\))

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

Example: If \(P = \{a, e, i, o, u\}\), then \(n(P) = 5\).

Note: For the Empty Set, \(n(\emptyset) = 0\).

The Principle of Inclusion-Exclusion (for two sets)

When calculating the size of a union, we must ensure we don't count the overlapping elements (the intersection) twice.

The fundamental formula for two sets is:
\[n(A \cup B) = n(A) + n(B) - n(A \cap B)\]

Analogy: Imagine counting students who play Football (A) and students who play Basketball (B). If you add \(n(A) + n(B)\), you have counted the players who play BOTH sports twice. You must subtract \(n(A \cap B)\) once to correct the count.

Solving Problems Using Cardinality

Many exam questions involve real-world scenarios requiring you to fill in a Venn diagram based on given data.

Process for Cardinality Problems (especially survey data):

1. Draw the diagram: Sketch the universal set rectangle and the necessary overlapping circles (usually 2 or 3).
2. Start in the deepest intersection: If you have 3 sets (A, B, C), start by filling in the value for \(n(A \cap B \cap C)\). This is the only place the element count can go.
3. Work outwards: Use the intersection totals given to find the number that belongs ONLY to that section.
   Example: If you know \(n(A \cap B) = 10\), and the center region (\(A \cap B \cap C\)) is 3, then the number who do A and B only is \(10 - 3 = 7\).
4. Fill in the "Only" regions: Use the total cardinality of each set (\(n(A)\), \(n(B)\), etc.) and subtract all the overlapping numbers you have already placed.
5. Find the remainder: Add up all the numbers placed inside the circles and subtract this total from the Universal Set total (\(n(\mathcal{E})\)) to find the count of elements outside all sets.

Did you know? Set theory was formalized in the late 19th century by mathematician Georg Cantor, and it completely changed how mathematicians think about numbers and infinity!

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Chapter Summary

You've covered the core concepts of Sets! Remember the key language:

• Sets are groups, elements are the members.
• The Universal Set \(\mathcal{E}\) is the container.
• Intersection (\(\cap\)) is "AND", Union (\(\cup\)) is "OR".
• Venn Diagrams are your best friend for visualization and problem-solving.
• Cardinality \(n(A)\) counts the elements. Always use the Inclusion-Exclusion formula when dealing with overlapping sets to avoid double counting!

Keep practicing filling in those Venn diagrams—it’s the fastest way to master problem-solving in this chapter. You’ve got this!