Sets, Subsets, Operations, Venn Diagrams & Applications

TAKEAWAYS:

• A set is a well-defined collection of distinct objects (e.g., {1, 2, 3} or {apple, banana}).
• Subsets (⊆) and proper subsets (⊂) show "part of" relationships between sets.
• Set operations (∪, ∩, A′, A–B) combine or compare sets using Venn diagrams.
• Universal set (U) and empty set (∅) are foundational for all set problems.
• Applications include business data classification, logical reasoning, and probability.
• NEB exams test Venn diagram shading, word-to-set conversion, and operation calculations.

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1. What is a Set?

A set is a well-defined collection of distinct objects, called elements or members.

• Well-defined: Clear rules for membership (e.g., "numbers less than 5" is a set; "tall people" is not).
• Distinct: No repeats (e.g., {1, 2, 2, 3} = {1, 2, 3}).

How to Write Sets

• Roster form: List elements inside curly braces. Example: $A=\{2,4,6,8\}$ (set of even numbers ≤ 8).
• Set-builder form: Describe the rule. Example: $B=\{x\mid x\text{ is a prime number }<10\}=\{2,3,5,7\}$.

Types of Sets

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Example 1.1: Write the following in roster form:

1. The set of vowels in "mathematics".
2. The set of months with 31 days.

Solution:

1. $V=\{a,e,i\}$ (only distinct vowels).
2. $M=\{\text{January, March, May, July, August, October, December}\}$.

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2. Subsets, Proper Subsets, and Equal Sets

Subset (⊆)

• If every element of set $A$ is also in set $B$, then $A\subseteq B$.
• Example: $A=\{1,2\}$, $B=\{1,2,3\}$. Here, $A\subseteq B$.

Proper Subset (⊂)

• If $A\subseteq B$ and $A\ne B$, then $A$ is a proper subset of $B$ ($A\subset B$).
• Example: $A=\{1,2\}\subset \{1,2,3\}$.

Equal Sets (=)

• Two sets are equal if they have exactly the same elements.
• Example: $P=\{x\mid x\text{ is a color in the Nepali flag}\}=\{blue,red\}$.

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Example 2.1: Given $X=\{a,b,c\}$, $Y=\{a,b\}$, $Z=\{a,b,c,d\}$:

1. Is $Y\subseteq X$? Yes (all elements of $Y$ are in $X$).
2. Is $X\subset Z$? Yes (proper subset).
3. Are $X$ and $Z$ equal? No ($Z$ has an extra element $d$).

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3. Set Operations

Union (∪)

• Combines all elements from both sets.
• Formula: $A\cup B=\{x\mid x\in A\text{ or }x\in B\}$.
• Example: $A=\{1,2\}$, $B=\{2,3\}$. $A\cup B=\{1,2,3\}$.

Intersection (∩)

• Elements common to both sets.
• Formula: $A\cap B=\{x\mid x\in A\text{ and }x\in B\}$.
• Example: $A\cap B=\{2\}$.

Complement (A′ or U – A)

• Elements in the universal set $U$ not in $A$.
• Formula: $A^{\prime }=U-A$.
• Example: If $U=\{1,2,3,4\}$, $A=\{1,2\}$, then $A^{\prime }=\{3,4\}$.

Difference (A – B)

• Elements in $A$ but not in $B$.
• Formula: $A-B=\{x\mid x\in A\text{ and }x\notin B\}$.
• Example: $A-B=\{1\}$.

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Example 3.1: Let $U=\{1,2,3,4,5\}$, $A=\{1,2,3\}$, $B=\{3,4\}$. Find:

1. $A\cup B$
2. $A\cap B$
3. $A^{\prime }$
4. $B-A$

Solution:

1. $A\cup B=\{1,2,3,4\}$.
2. $A\cap B=\{3\}$.
3. $A^{\prime }=U-A=\{4,5\}$.
4. $B-A=\{4\}$ (only 4 is in $B$ but not in $A$).

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4. Venn Diagrams and Shading

Venn diagrams visually represent sets and operations. Rules for Shading:

1. Union (A ∪ B): Shade both circles.
2. Intersection (A ∩ B): Shade the overlapping area.
3. Complement (A′): Shade the universal rectangle except circle $A$.
4. Difference (A – B): Shade only the part of $A$ not overlapping with $B$.

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Example 4.1: Shade the region for $(A\cup B{)}^{\prime }$ in the Venn diagram below. Solution:

• $(A\cup B{)}^{\prime }$ means everything outside both $A$ and $B$.
• Shade the area in $U$ but not in $A$ or $B$.

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5. Applications of Sets in Business

Sets help organize and analyze data in business:

1. Customer Segmentation:
   • Classify customers by age, location, or purchase history.
   • Example: $A=\{\text{Customers who bought Product X}\}$, $B=\{\text{Customers who bought Product Y}\}$.
   • $A\cap B$ = Customers who bought both products (target for cross-selling).

2. Inventory Management:

   • $U=\{\text{All products}\}$, $A=\{\text{Products in stock}\}$, $B=\{\text{Products sold out}\}$.
   • $A^{\prime }$ = Products not in stock (need to reorder).

3. Logical Reasoning:

   • Used in decision-making (e.g., "If $A$ happens, then $B$ must follow").

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Example 5.1: A company surveys 50 employees:

• $A=\{\text{Employees who use laptop}\}=30$.
• $B=\{\text{Employees who use smartphone}\}=25$.
• $A\cap B=15$ (use both). Find:

1. Employees who use only laptop.
2. Employees who use neither.

Solution:

1. Only laptop = $A-B=30-15=15$.
2. Neither = $U-(A\cup B)$.
   • $A\cup B=30+25-15=40$.
   • Neither = $50-40=10$.

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6. Important Formulas and Laws

Example 6.1: Verify De Morgan’s Law for $A=\{1,2\}$, $B=\{2,3\}$, $U=\{1,2,3,4\}$. Solution:

1. $A\cup B=\{1,2,3\}$. $(A\cup B{)}^{\prime }=U-\{1,2,3\}=\{4\}$.
2. $A^{\prime }=\{3,4\}$, $B^{\prime }=\{1,4\}$. $A^{\prime }\cap B^{\prime }=\{4\}$. Thus, $(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$ ✓.

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Exam Tip: How to Score Full Marks in NEB Exams

1. Understand the Question:

   • Convert word problems into sets (e.g., "students who passed Maths" → $A$, "students who passed English" → $B$).

2. Draw Venn Diagrams:

   • Always draw diagrams for union, intersection, and complement questions.
   • Label all regions clearly.

3. Use Formulas Correctly:

   • For $n(A\cup B)$, use $n(A)+n(B)-n(A\cap B)$.
   • For complements, remember $n(A^{\prime })=n(U)-n(A)$.

4. Check Your Work:

   • Verify with examples (e.g., if $A\subseteq B$, then $A\cup B=B$).

5. Common Mistakes to Avoid:

   • Forgetting to subtract the intersection in $n(A\cup B)$.
   • Misplacing complements (e.g., $(A\cup B{)}^{\prime }\ne A^{\prime }\cup B^{\prime }$).
   • Not considering the universal set $U$ in complement problems.

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NEB-Style Practice Questions

Short Answer (5 marks each)

1. Define: a) Subset b) Universal set c) Difference of sets $A-B$

2. Given $U=\{1,2,3,4,5,6\}$, $A=\{1,2,3\}$, $B=\{3,4,5\}$: a) Find $A\cup B$. b) Find $A^{\prime }\cap B$. c) Draw a Venn diagram and shade $(A\cap B{)}^{\prime }$.

3. In a class of 40 students:

   • 25 study Maths ($M$).
   • 20 study English ($E$).
   • 10 study both. Find: a) Students studying only Maths. b) Students studying neither.

Long Answer (10 marks)

4. A survey of 100 people found:
   • 60 like tea ($T$).
   • 50 like coffee ($C$).
   • 30 like both. a) How many like only tea? b) How many like neither? c) Draw a Venn diagram and label all regions with numbers.

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Answers: 1. a) $A\subseteq B$ if every element of $A$ is in $B$. b) The set containing all elements under consideration. c) $A-B=\{x\mid x\in A\text{ and }x\notin B\}$.

2. a) $A\cup B=\{1,2,3,4,5\}$. b) $A^{\prime }=\{4,5,6\}$, so $A^{\prime }\cap B=\{4,5\}$. c) Shade everything except the overlapping part of $A$ and $B$.

3. a) Only Maths = $n(M)-n(M\cap E)=25-10=15$. b) Neither = $40-(25+20-10)=5$.

4. a) Only tea = $60-30=30$. b) Neither = $100-(60+50-30)=20$. c) Venn diagram:

   • $T$ only: 30
   • $C$ only: 20
   • Both: 30
   • Neither: 20.