Examples of Set Operations and Counting Problems

A well-defined collection of distinct objects called elements or members.

Learning Outcomes:

1. Understand set notations.
2. Solve counting problems using Venn diagrams.

Here is a detailed guide covering set notations and solving counting problems using Venn diagrams with 10 examples, including diagrams and emojis to enhance clarity.

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1️⃣ Understanding Set Notations

Basic Set Notations and Symbols:

• Set: Denoted by curly braces ${}$. Example: $A = {1,2,3}$
• Element of a Set: $x \in A$ means $x$ is in set $A$.
• Not an Element: $x \notin A$ means $x$ is not in set $A$.
• Subset: $A \subseteq B$ means every element of $A$ is in $B$.
• Proper Subset: $A \subset B$ means $A$ is subset but not equal to $B$.
• Empty Set: $\emptyset$, set with no elements.
• Universal Set: $U$, all elements under consideration.
• Union: $A \cup B = {x | x \in A \text{ or } x \in B}$
• Intersection: $A \cap B = {x | x \in A \text{ and } x \in B}$
• Set Difference: $A - B = {x | x \in A \text{ and } x \notin B}$
• Complement: $A^c = U - A$

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Visual Diagram of Set Operations:

🔵 ∪, 🔴 ∩, areas represent unions and intersections.

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2️⃣ Solve Counting Problems Using Venn Diagrams

Counting problems involving overlapping sets can be solved efficiently by Venn diagrams.

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Example 1: Basic Two-Set Problem

• 40 students study Math, 30 study English, and 10 study both.
• How many students study at least one?

$$ |M \cup E| = |M| + |E| - |M \cap E| = 40 + 30 - 10 = 60 $$

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Example 2: Three-Set Problem

• 50 students play football, 40 play basketball, 30 play volleyball, 20 play football & basketball,
• 15 play basketball & volleyball, 10 play football & volleyball, 5 play all three.
• Find how many play at least one sport.

$$ |F \cup B \cup V| = |F| + |B| + |V| - |F \cap B| - |B \cap V| - |F \cap V| + |F \cap B \cap V| $$

Calculate:

$$ 50 + 40 + 30 - 20 - 15 - 10 + 5 = 80 $$

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Example 3: Students in Subjects

• 60 students: 35 study Physics, 30 Chemistry, 20 both.
• How many study neither?

Assuming total = 60:

$$ |P \cup C| = 35 + 30 - 20 = 45 $$

Neither = $60 - 45 = 15$

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Example 4: Overlapping Preferences

• Out of 100 people, 70 like tea, 50 like coffee, 30 like both.
• How many like either tea or coffee but not both?

$$ (70 - 30) + (50 - 30) = 60 $$

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Example 5: Book Readers

• 100 people reading books: 70 read fiction, 40 non-fiction, 25 both.
• Find how many read only fiction.

Only fiction = $70 - 25 = 45$

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Example 6: Survey with Three Choices

• In a group, liked: A = 80, B = 65, C = 75, A&B = 45, B&C = 35, A&C = 30, All three = 25.
• Find how many liked none if total is 150.

At least one:

$$ 80 + 65 + 75 - 45 - 35 - 30 + 25 = 135 $$

None = $150 - 135 = 15$

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Example 7: Sports Participation

• 70 students play Cricket, 60 Football, 50 Hockey, 30 Cricket & Football,
• 20 Football & Hockey, 15 Cricket & Hockey, 5 all three.
• How many play only one sport?

Calculate only one set:

$$ \begin{cases} C \text{ only} = 70 - 30 - 15 + 5 = 30 \\ F \text{ only} = 60 - 30 - 20 + 5 = 15 \\ H \text{ only} = 50 - 15 - 20 + 5 = 20 \\ \end{cases} $$

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Example 8: Analyzing Survey Data

• 120 people surveyed: 50 like apples, 70 like bananas, 30 like both.
• How many like only one fruit?

$$ (50 - 30) + (70 - 30) = 60 $$

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Example 9: Multi-Category Venn

• In a town: 200 people use Internet, 150 use TV, 100 use Newspaper,
• Internet & TV = 80, TV & Newspaper = 60, Internet & Newspaper = 40,
• All three = 20, total population 300.
• How many use none?

Calculate people using any:

$$ 200 + 150 + 100 - 80 - 60 - 40 + 20 = 290 $$

None = $300 - 290 = 10$

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Example 10: Students with Multiple Skills

• 45 students know Java, 30 know Python, 20 know both.
• Total students = 60.
• How many know neither?

$$ |J \cup P| = 45 + 30 - 20 = 55 $$

Neither = $60 - 55 = 5$

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These examples utilize Venn diagrams and formulas to solve counting problems involving sets effectively.

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Emojis summary:

• 🔷 Set notation
• 🔵 Circle for sets in diagrams
• 🧮 Counting elements
• 🎯 Intersection and union
• ❌ Excluding elements

These illustrations and examples offer solid practice in set theory and counting with Venn diagrams.

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Exercise Questions 🧠

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