Sets

1. Overview

Sets are a fundamental concept in mathematics used to group objects or numbers with shared properties. This topic teaches you how to define sets, perform operations on them (like union and intersection), and visually represent them using Venn diagrams. Mastering sets is crucial for solving logic problems, understanding probability, and organizing data effectively. The IGCSE exam requires you to be fluent in set notation and able to apply these concepts to solve problems.

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Key Definitions

• Set: A collection of distinct objects or numbers, usually denoted by a capital letter (e.g., $A$).
• Element ($\in$): An individual item or member within a set. For example, if $A = {1, 2, 3}$, then $2 \in A$.
• Universal Set ($\mathscr{E}$ or $\xi$): The set containing all possible elements being considered in a specific problem. It's the "universe" your sets live in.
• Empty Set ($\emptyset$ or ${}$): A set that contains no elements.
• Complement ($A'$): All elements in the Universal Set that are not in set $A$.
• Intersection ($A \cap B$): Elements that belong to both set $A$ and set $B$.
• Union ($A \cup B$): Elements that belong to set $A$, or set $B$, or both.
• Subset ($A \subseteq B$): Every element in $A$ is also an element of $B$.
• Proper Subset ($A \subset B$): Every element in $A$ is in $B$, but $B$ has at least one element not in $A$.
• Cardinality ($n(A)$): The number of elements in set $A$.

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Core Content

Listing Elements and Cardinality

When listing a set, use curly brackets ${}$ and separate elements with commas.

• Example: $\xi = {x : x \text{ is an integer, } 1 \leq x \leq 10}$
• $A = { \text{Prime numbers less than 10} }$
• Step 1: Identify prime numbers: $2, 3, 5, 7$.
• Step 2: List the set: $A = {2, 3, 5, 7}$.
• Step 3: Count the elements for $n(A)$: $n(A) = 4$.

Venn Diagrams

Venn diagrams represent sets visually. Elements are written inside circles; elements not in specific sets are written in the surrounding rectangle (the Universal Set).

Worked Example 1 — Intersection and Union

Given $\xi = {1, 2, 3, 4, 5, 6}$, $A = {1, 2, 3, 4}$ and $B = {3, 4, 5, 6}$.

1. Find $A \cap B$: Look for common elements.
   • $3$ and $4$ are in both.
   • Result: $A \cap B = {3, 4}$.
2. Find $A \cup B$: Combine all elements from both, without repeats.
   • $1, 2, 3, 4 + 5, 6$.
   • Result: $A \cup B = {1, 2, 3, 4, 5, 6}$.

Worked example 2 — Finding the complement

Question: Given the universal set $\mathscr{E} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ and the set $A = {2, 4, 6, 8}$, find $A'$.

Solution:

• Step 1: Identify the elements in the universal set $\mathscr{E}$.
  • $\mathscr{E} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$
  • Reason: This is given in the problem statement.
• Step 2: Identify the elements in set $A$.
  • $A = {2, 4, 6, 8}$
  • Reason: This is given in the problem statement.
• Step 3: Determine the elements that are in $\mathscr{E}$ but not in $A$.
  • $A' = {1, 3, 5, 7, 9, 10}$
  • Reason: These are the elements present in the universal set but not in set A.

Final Answer: $A' = {1, 3, 5, 7, 9, 10}$

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Extended Content (Extended Curriculum Only)

Advanced Venn Diagram Shading

Students are often asked to shade specific regions.

• $P \cup Q'$: This means "Everything in $P$ OR everything NOT in $Q$."
• $(A \cap B)'$: This means "The complement of the intersection" (everything except the middle overlap).

Worked Example 3 — Solving for $x$ in Venn Diagrams

Question: In a survey of 80 people, 45 like coffee ($C$), 30 like tea ($T$), and 10 like neither. Let $x$ represent the number of people who like both coffee and tea. Find the value of $x$.

• Step 1: Express "Only $C$" and "Only $T$" in terms of $x$.
  • $n(C \text{ only}) = 45 - x$
  • $n(T \text{ only}) = 30 - x$
  • Reason: Subtract the intersection from the total number in each set.
• Step 2: Create an equation where the sum of all regions equals the total.
  • $(45 - x) + x + (30 - x) + 10 = 80$
  • Reason: The sum of those who like only coffee, only tea, both, and neither must equal the total number of people surveyed.
• Step 3: Simplify and solve.
  • $45 - x + x + 30 - x + 10 = 80$
  • Reason: Expanding the equation.
  • $85 - x = 80$
  • Reason: Combining like terms.
  • $-x = 80 - 85$
  • Reason: Subtracting 85 from both sides.
  • $-x = -5$
  • Reason: Simplifying.
  • $x = 5$
  • Reason: Multiplying both sides by -1.
• Final Answer: $\boxed{5}$ people like both coffee and tea.

Interpreting Set Statements

Equations like $n(F) = 3 \times n(V)$ describe the relationship between the entirety of the sets.

• If $n(V) = 10$, then the entire circle $V$ must contain 10 elements.
• If the intersection is $2$, then "$V$ only" is $10 - 2 = 8$.

Worked Example 4 — Using set statements to find cardinality

Question: Given that $n(\mathscr{E}) = 60$, $n(A) = 2n(B)$, $n(A \cap B) = 5$, and $n(A \cup B) = 40$, find $n(B)$.

Solution:

• Step 1: Use the Inclusion-Exclusion Principle: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$
  • $40 = n(A) + n(B) - 5$
  • Reason: Substituting the given values.
• Step 2: Substitute $n(A) = 2n(B)$ into the equation.
  • $40 = 2n(B) + n(B) - 5$
  • Reason: Replacing $n(A)$ with its equivalent expression.
• Step 3: Simplify and solve for $n(B)$.
  • $40 = 3n(B) - 5$
  • Reason: Combining like terms.
  • $45 = 3n(B)$
  • Reason: Adding 5 to both sides.
  • $n(B) = \frac{45}{3}$
  • Reason: Dividing both sides by 3.
  • $n(B) = 15$
  • Reason: Simplifying.

Final Answer: $\boxed{n(B) = 15}$

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Key Equations

Inclusion-Exclusion Principle (MEMORIZE):

$\qquad \bf n(A \cup B) = n(A) + n(B) - n(A \cap B)$

• $n(A \cup B)$: Total number in either set.
• $n(A), n(B)$: Total number in each set circle.
• $n(A \cap B)$: The overlap (subtracted so it isn't counted twice).

Complement Rule (MEMORIZE):

$\qquad \bf n(A) + n(A') = n(\mathscr{E})$

• $n(\mathscr{E})$: Total number of elements in the Universal Set.

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Common Mistakes to Avoid

• ❌ Wrong: Placing the value of $n(A)$ directly into the "Only $A$" region of a Venn diagram without considering the intersection. For example, if $n(A) = 30$ and $n(A \cap B) = 10$, incorrectly putting 30 in the "Only A" region.
  • ✓ Right: Subtract the intersection from the total set value to find the "Only $A$" region: $n(A) - n(A \cap B)$. In the example above, "Only A" would be $30 - 10 = 20$.
• ❌ Wrong: Failing to account for elements outside of sets $A$ and $B$ (i.e., elements in neither set) when calculating the total number of elements in the universal set.
  • ✓ Right: Ensure your equation includes all regions: $n(\text{Only } A) + n(\text{Only } B) + n(A \cap B) + n(\text{Neither}) = n(\mathscr{E})$.
• ❌ Wrong: When shading a Venn diagram for $P \cup Q'$, only shading the circle representing set $P$.
  • ✓ Right: $P \cup Q'$ requires shading the entire circle $P$ plus everything outside of circle $Q$.

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Exam Tips

• Command Words:
  • "List the elements...": Use ${ }$ brackets.
  • "Find $n(A)$": Give a single number (count the elements).
  • "Shade the region...": Use clear, diagonal lines.
• Calculator Tip: While set theory is about logic, use your calculator to double-check all arithmetic calculations in Venn diagram problems, especially when dealing with larger numbers.
• Marks Alert: Show your working clearly, especially the subtraction of the intersection when finding the number of elements in "Only A" or "Only B". You will lose marks for a correct answer without supporting steps like ($15 - x$).
• Formulas: The Inclusion-Exclusion principle is not provided on the formula sheet; you must memorize it.