Class 8
Mathematics
Federal Board of Intermediate and Secondary Education (FBISE) (FBISE)
📚 Model Textbook of Mathematics Grade 8 (FBISE / NBF)
Class 8 Mathematics - Ch 5: Sets Mastery Guide (FBISE)
📖 Chapter 5: Sets
📅 Updated: Sep 13, 2026
Teacher & Student Roadmap
Grade 8 Mathematics • FBISE / National Curriculum (NBF)
Instructional Guide: Unit 05 Sets (سیٹ)
Target Student Learning Outcomes (SLOs)
- Express sets seamlessly in Descriptive, Tabular (Roster), and Set-Builder notations.
- Construct Power Sets $P(A)$ containing $2^n$ subsets and distinguish between proper ($\subset$) and improper ($\subseteq$) subsets.
- Execute core set operations: Union ($\cup$), Intersection ($\cap$), Difference ($-$), and Complement ($A' = U - A$).
- Verify fundamental algebraic set laws: Commutative, Associative, and Distributive Laws.
- Prove and demonstrate De Morgan’s Laws: $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$.
- Construct and shade Venn Diagrams for Disjoint, Overlapping, and Subset configurations, and solve real-world survey word problems.
Common Student Misconceptions & Traps
- Power Set of Empty Set: Thinking $P(\emptyset) = \emptyset$. The power set of an empty set is NOT empty; it contains one element: $P(\emptyset) = \{\emptyset\}$, so $n(P(\emptyset)) = 2^0 = 1$.
- De Morgan's Sign Flip: Forgetting to flip the operator inside the complement: $(A \cup B)' \to A' \cap B'$ (Union becomes Intersection), and $(A \cap B)' \to A' \cup B'$ (Intersection becomes Union).
- Difference Non-Commutativity: Assuming $A - B = B - A$. In sets, $A - B$ contains elements in $A$ not in $B$, whereas $B - A$ contains elements in $B$ not in $A$.
- Proper vs. Improper Subsets: Every set has exactly ONE improper subset (itself). Total proper subsets $= 2^n - 1$.
- Set Builder Symbols: Confusing $\land$ (AND $\implies$ Intersection) with $\lor$ (OR $\implies$ Union).
3-Step Concept Mastery Strategy
- Step 1: Translate to Tabular Listing: Convert descriptive or set-builder statements into explicit comma-separated elements within braces $\{ \}$.
- Step 2: Apply Set Algebraic Operations: Compute brackets first (LHS & RHS independently), using formal membership rules.
- Step 3: Venn Diagram Cross-Verification: Sketch the universal rectangle $U$, draw circles for sets, place elements in appropriate partitioned regions, and shade corresponding areas.
Curriculum Alignment: This unit strictly follows the Single National Curriculum (SNC) / FBISE Grade 8 Mathematics textbook published by the National Book Foundation (NBF). All set builder predicates, power sets, Venn diagram shading schemes, and 100% solved exercises are fully elaborated with step-by-step scoring breakdown.
Chapter 5: Sets (سیٹ)
A set is the fundamental building block of modern mathematics and computer science. Defined by German mathematician Georg Cantor as a well-defined collection of distinct objects, set theory provides the logical foundation for database searches, boolean logic, probability, and survey analytics.
💡 Kid-Friendly Tips for Success
- No Duplicates Allowed: A set never lists the same item twice: $\{1, 1, 2\} = \{1, 2\}$.
- Order Does Not Matter: $\{a, b, c\}$ is identical to $\{c, a, b\}$.
- Subset Counting Rule ($2^n$): A set with $n$ elements has $2^n$ total subsets and $2^n - 1$ proper subsets.
- De Morgan Flip: The prime outside flips the inside: $\cup \leftrightarrow \cap$.
- Complement Means "NOT in A": $A' = U - A$ (everything in universe $U$ except circle $A$).
🌍 Real-World Connections
- Search Engines & Databases: Google search uses Set Union (OR) and Set Intersection (AND) to retrieve web pages.
- Social Media Filters: Finding mutual friends on Facebook/Instagram is finding the Intersection ($A \cap B$) of two friend sets.
- School Survey Analytics: Determining students enrolled in both Biology and Computer Science using Venn diagrams.
- Library & Digital Catalogs: Categorizing books by genre, author, and language into disjoint or overlapping sets.
Visual Concept 1: Three Forms of Set Notation & Power Set Binary Lattice ($2^n$)
Visual Concept 2: Core Set Operations & Venn Diagram Shading Patterns
Visual Concept 3: De Morgan’s Laws & Distributive Laws Dualities
Section-by-Section Theoretical Mastery
1. Sets and Standard Number Sets
A set is a well-defined collection of distinct objects. Standard mathematical number sets include:
$\mathbb{N}$ (Natural): $\{1, 2, 3, 4, \dots\}$
$\mathbb{W}$ (Whole): $\{0, 1, 2, 3, 4, \dots\}$
$\mathbb{Z}$ (Integers): $\{0, \pm 1, \pm 2, \pm 3, \dots\}$
$\mathbb{E}$ (Even): $\{0, \pm 2, \pm 4, \pm 6, \dots\}$
$\mathbb{O}$ (Odd): $\{\pm 1, \pm 3, \pm 5, \dots\}$
$\mathbb{P}$ (Prime): $\{2, 3, 5, 7, 11, 13, \dots\}$
2. Subsets & Power Sets $P(A)$
- Subset ($A \subseteq B$): Every element of set $A$ is also an element of set $B$.
- Proper Subset ($A \subset B$): Set $A$ is contained in $B$ and $A \neq B$ (at least one element of $B$ is not in $A$).
- Improper Subset: Any set $A$ is an improper subset of itself ($A \subseteq A$).
- Power Set $P(A)$: The set consisting of all possible subsets of $A$. If $n(A) = n$, then:
$$n(P(A)) = 2^n, \quad \text{Proper subsets count} = 2^n - 1$$
3. Algebraic Laws of Sets
Commutative Laws:
$A \cup B = B \cup A$
$A \cap B = B \cap A$
$A \cup B = B \cup A$
$A \cap B = B \cap A$
Associative Laws:
$(A \cup B) \cup C = A \cup (B \cup C)$
$(A \cap B) \cap C = A \cap (B \cap C)$
$(A \cup B) \cup C = A \cup (B \cup C)$
$(A \cap B) \cap C = A \cap (B \cap C)$
Distributive Laws:
$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
4. De Morgan's Laws & Venn Diagrams
British mathematician Augustus De Morgan formulated two laws connecting complement, union, and intersection: $$(A \cup B)' = A' \cap B' \quad \text{and} \quad (A \cap B)' = A' \cup B'$$ These laws are graphically verified by showing that the shaded Venn diagram region of the LHS matches identically with the RHS.
Exercise 5.1 • Set Notations, Subsets & Power Sets
Q1. Write the following in set builder form:
(i) $\{1, 2, 3, \dots\} = \mathbf{\{x : x \in \mathbb{N}\}}$
(ii) $\{1, 2, 3, \dots, 10\} = \mathbf{\{x : x \in \mathbb{N} \land x \le 10\}}$
(iii) $\{\pm 1, \pm 2, \pm 3, \pm 4\} = \mathbf{\{x : x \in \mathbb{Z} \land -4 \le x \le 4 \land x \neq 0\}}$
(iv) $\{-1, -2, -3, -4, -5\} = \mathbf{\{x : x \in \mathbb{Z}^- \land -5 \le x \le -1\}}$
(v) $\{0, 4, 8, 12, 16, 20\} = \mathbf{\{x : x = 4n \land n \in \mathbb{W} \land n \le 5\}}$
(vi) $\{12, 14, 16, \dots, 100\} = \mathbf{\{x : x \in \mathbb{E}^+ \land 12 \le x \le 100\}}$
(vii) $\{1, 2, 3, 6, 9, 18\} = \mathbf{\{x : x \text{ is a factor of } 18\}}$
(viii) Set of whole numbers $= \mathbf{\{x : x \in \mathbb{W}\}}$
(ix) Set of integers between $-10$ and $3 = \mathbf{\{x : x \in \mathbb{Z} \land -10 < x < 3\}}$
(x) Set of prime numbers between $5$ and $20 = \mathbf{\{x : x \in \mathbb{P} \land 5 < x < 20\}}$
(xi) Set of all districts of Pakistan $= \mathbf{\{x : x \text{ is a district of Pakistan}\}}$
(xii) Set of integers between 1 and $2 = \mathbf{\{x : x \in \mathbb{Z} \land 1 < x < 2\}}$
(xiii) Set of all Pakistani citizens $= \mathbf{\{x : x \text{ is a citizen of Pakistan}\}}$
(xiv) Set of odd numbers from $-1$ to $-13 = \mathbf{\{x : x \in \mathbb{O}^- \land -13 \le x \le -1\}}$
(xv) Set of natural numbers greater than $10 = \mathbf{\{x : x \in \mathbb{N} \land x > 10\}}$
(ii) $\{1, 2, 3, \dots, 10\} = \mathbf{\{x : x \in \mathbb{N} \land x \le 10\}}$
(iii) $\{\pm 1, \pm 2, \pm 3, \pm 4\} = \mathbf{\{x : x \in \mathbb{Z} \land -4 \le x \le 4 \land x \neq 0\}}$
(iv) $\{-1, -2, -3, -4, -5\} = \mathbf{\{x : x \in \mathbb{Z}^- \land -5 \le x \le -1\}}$
(v) $\{0, 4, 8, 12, 16, 20\} = \mathbf{\{x : x = 4n \land n \in \mathbb{W} \land n \le 5\}}$
(vi) $\{12, 14, 16, \dots, 100\} = \mathbf{\{x : x \in \mathbb{E}^+ \land 12 \le x \le 100\}}$
(vii) $\{1, 2, 3, 6, 9, 18\} = \mathbf{\{x : x \text{ is a factor of } 18\}}$
(viii) Set of whole numbers $= \mathbf{\{x : x \in \mathbb{W}\}}$
(ix) Set of integers between $-10$ and $3 = \mathbf{\{x : x \in \mathbb{Z} \land -10 < x < 3\}}$
(x) Set of prime numbers between $5$ and $20 = \mathbf{\{x : x \in \mathbb{P} \land 5 < x < 20\}}$
(xi) Set of all districts of Pakistan $= \mathbf{\{x : x \text{ is a district of Pakistan}\}}$
(xii) Set of integers between 1 and $2 = \mathbf{\{x : x \in \mathbb{Z} \land 1 < x < 2\}}$
(xiii) Set of all Pakistani citizens $= \mathbf{\{x : x \text{ is a citizen of Pakistan}\}}$
(xiv) Set of odd numbers from $-1$ to $-13 = \mathbf{\{x : x \in \mathbb{O}^- \land -13 \le x \le -1\}}$
(xv) Set of natural numbers greater than $10 = \mathbf{\{x : x \in \mathbb{N} \land x > 10\}}$
Q2. Write the following sets in tabular and descriptive form:
(i) $\{x : x \in \mathbb{N} \land x = 5\}$: Tabular: $\mathbf{\{5\}}$, Descriptive: Set containing only the natural number 5.
(ii) $\{x : x \in \mathbb{N} \land x > 8\}$: Tabular: $\mathbf{\{9, 10, 11, \dots\}}$, Descriptive: Set of natural numbers greater than 8.
(iii) $\{y : y \in \mathbb{W} \land y \le 6\}$: Tabular: $\mathbf{\{0, 1, 2, 3, 4, 5, 6\}}$, Descriptive: Set of whole numbers up to 6.
(iv) $\{x : x \in \mathbb{Z} \land -3 \le x \le 3\}$: Tabular: $\mathbf{\{-3, -2, -1, 0, 1, 2, 3\}}$, Descriptive: Set of integers from $-3$ to $3$.
(v) $\{y : y \in \mathbb{O} \land 1 < y < 16\}$: Tabular: $\mathbf{\{3, 5, 7, 9, 11, 13, 15\}}$, Descriptive: Set of odd numbers between 1 and 16.
(vi) $\{x : x \in \mathbb{E} \land x > 2\}$: Tabular: $\mathbf{\{4, 6, 8, \dots\}}$, Descriptive: Set of positive even numbers greater than 2.
(vii) $\{y : y \in \mathbb{E} \land -10 \le y \le 0\}$: Tabular: $\mathbf{\{-10, -8, -6, -4, -2, 0\}}$, Descriptive: Set of non-positive even numbers from $-10$ to $0$.
(viii) $\{3x : x \in \mathbb{N} \land x < 6\}$: Tabular: $\mathbf{\{3, 6, 9, 12, 15\}}$, Descriptive: Set of first five positive multiples of 3.
(ix) $\{x : x \text{ is textbook of your class}\}$: Tabular: $\mathbf{\{\text{Math, Science, English, Urdu, Islamiat, Social Studies, Computer}\}}$, Descriptive: Set of Grade 8 textbooks.
(x) $\{y : y \in \mathbb{W} \land 2 < y < 3\}$: Tabular: $\mathbf{\emptyset \text{ or } \{\}}$, Descriptive: Empty set of whole numbers between 2 and 3.
(xi) $\{x : x \text{ is a basic principle of Islam}\}$: Tabular: $\mathbf{\{\text{Tauheed, Salat, Zakat, Sawm, Hajj}\}}$, Descriptive: Set of five pillars of Islam.
(xii) $\{x : x \text{ is a multiple of } 5\}$: Tabular: $\mathbf{\{5, 10, 15, \dots\}}$, Descriptive: Set of positive multiples of 5.
(xiii) $\{y : y \text{ is a factor of } 24\}$: Tabular: $\mathbf{\{1, 2, 3, 4, 6, 8, 12, 24\}}$, Descriptive: Set of positive factors of 24.
(ii) $\{x : x \in \mathbb{N} \land x > 8\}$: Tabular: $\mathbf{\{9, 10, 11, \dots\}}$, Descriptive: Set of natural numbers greater than 8.
(iii) $\{y : y \in \mathbb{W} \land y \le 6\}$: Tabular: $\mathbf{\{0, 1, 2, 3, 4, 5, 6\}}$, Descriptive: Set of whole numbers up to 6.
(iv) $\{x : x \in \mathbb{Z} \land -3 \le x \le 3\}$: Tabular: $\mathbf{\{-3, -2, -1, 0, 1, 2, 3\}}$, Descriptive: Set of integers from $-3$ to $3$.
(v) $\{y : y \in \mathbb{O} \land 1 < y < 16\}$: Tabular: $\mathbf{\{3, 5, 7, 9, 11, 13, 15\}}$, Descriptive: Set of odd numbers between 1 and 16.
(vi) $\{x : x \in \mathbb{E} \land x > 2\}$: Tabular: $\mathbf{\{4, 6, 8, \dots\}}$, Descriptive: Set of positive even numbers greater than 2.
(vii) $\{y : y \in \mathbb{E} \land -10 \le y \le 0\}$: Tabular: $\mathbf{\{-10, -8, -6, -4, -2, 0\}}$, Descriptive: Set of non-positive even numbers from $-10$ to $0$.
(viii) $\{3x : x \in \mathbb{N} \land x < 6\}$: Tabular: $\mathbf{\{3, 6, 9, 12, 15\}}$, Descriptive: Set of first five positive multiples of 3.
(ix) $\{x : x \text{ is textbook of your class}\}$: Tabular: $\mathbf{\{\text{Math, Science, English, Urdu, Islamiat, Social Studies, Computer}\}}$, Descriptive: Set of Grade 8 textbooks.
(x) $\{y : y \in \mathbb{W} \land 2 < y < 3\}$: Tabular: $\mathbf{\emptyset \text{ or } \{\}}$, Descriptive: Empty set of whole numbers between 2 and 3.
(xi) $\{x : x \text{ is a basic principle of Islam}\}$: Tabular: $\mathbf{\{\text{Tauheed, Salat, Zakat, Sawm, Hajj}\}}$, Descriptive: Set of five pillars of Islam.
(xii) $\{x : x \text{ is a multiple of } 5\}$: Tabular: $\mathbf{\{5, 10, 15, \dots\}}$, Descriptive: Set of positive multiples of 5.
(xiii) $\{y : y \text{ is a factor of } 24\}$: Tabular: $\mathbf{\{1, 2, 3, 4, 6, 8, 12, 24\}}$, Descriptive: Set of positive factors of 24.
Q3. Write the number of elements in the power set of following sets ($2^n$):
(i) $\{1, 3, 5, 7, 9\}$: $n = 5 \implies 2^5 = \mathbf{32\text{ elements}}$.
(ii) $\{0, 1, 2, 3, 4, 5\}$: $n = 6 \implies 2^6 = \mathbf{64\text{ elements}}$.
(iii) $\emptyset$: $n = 0 \implies 2^0 = \mathbf{1\text{ element}}$.
(ii) $\{0, 1, 2, 3, 4, 5\}$: $n = 6 \implies 2^6 = \mathbf{64\text{ elements}}$.
(iii) $\emptyset$: $n = 0 \implies 2^0 = \mathbf{1\text{ element}}$.
Q4. Find the power set of the following sets:
(i) $A = \{a, b\}$ ($2^2 = 4$):
$$P(A) = \mathbf{\{\emptyset, \{a\}, \{b\}, \{a, b\}\}}$$
(ii) $B = \{0, 2, 4\}$ ($2^3 = 8$):
$$P(B) = \mathbf{\{\emptyset, \{0\}, \{2\}, \{4\}, \{0, 2\}, \{0, 4\}, \{2, 4\}, \{0, 2, 4\}\}}$$
(iii) $C = \{1, 2, 3, 4\}$ ($2^4 = 16$):
$$P(C) = \mathbf{\{\emptyset, \{1\}, \{2\}, \{3\}, \{4\}, \{1, 2\}, \{1, 3\}, \{1, 4\}, \{2, 3\}, \{2, 4\}, \{3, 4\}, \{1, 2, 3\}, \{1, 2, 4\}, \{1, 3, 4\}, \{2, 3, 4\}, \{1, 2, 3, 4\}\}}$$
(iv) $D = \{101\}$ ($2^1 = 2$):
$$P(D) = \mathbf{\{\emptyset, \{101\}\}}$$
(v) $E = \emptyset$ ($2^0 = 1$):
$$P(E) = \mathbf{\{\emptyset\}}$$
$$P(A) = \mathbf{\{\emptyset, \{a\}, \{b\}, \{a, b\}\}}$$
(ii) $B = \{0, 2, 4\}$ ($2^3 = 8$):
$$P(B) = \mathbf{\{\emptyset, \{0\}, \{2\}, \{4\}, \{0, 2\}, \{0, 4\}, \{2, 4\}, \{0, 2, 4\}\}}$$
(iii) $C = \{1, 2, 3, 4\}$ ($2^4 = 16$):
$$P(C) = \mathbf{\{\emptyset, \{1\}, \{2\}, \{3\}, \{4\}, \{1, 2\}, \{1, 3\}, \{1, 4\}, \{2, 3\}, \{2, 4\}, \{3, 4\}, \{1, 2, 3\}, \{1, 2, 4\}, \{1, 3, 4\}, \{2, 3, 4\}, \{1, 2, 3, 4\}\}}$$
(iv) $D = \{101\}$ ($2^1 = 2$):
$$P(D) = \mathbf{\{\emptyset, \{101\}\}}$$
(v) $E = \emptyset$ ($2^0 = 1$):
$$P(E) = \mathbf{\{\emptyset\}}$$
Q5. Write the power set of letters of the following words:
(i) $\text{FAN} = \{F, A, N\}$ ($2^3 = 8$):
$$P(\text{FAN}) = \mathbf{\{\emptyset, \{F\}, \{A\}, \{N\}, \{F, A\}, \{F, N\}, \{A, N\}, \{F, A, N\}\}}$$
(ii) $\text{AAQIB}$ (Distinct letters $= \{A, Q, I, B\}$, $n = 4$, $2^4 = 16$):
$$P(\text{AAQIB}) = \mathbf{\{\emptyset, \{A\}, \{Q\}, \{I\}, \{B\}, \{A, Q\}, \{A, I\}, \{A, B\}, \{Q, I\}, \{Q, B\}, \{I, B\}, \{A, Q, I\}, \{A, Q, B\}, \{A, I, B\}, \{Q, I, B\}, \{A, Q, I, B\}\}}$$
$$P(\text{FAN}) = \mathbf{\{\emptyset, \{F\}, \{A\}, \{N\}, \{F, A\}, \{F, N\}, \{A, N\}, \{F, A, N\}\}}$$
(ii) $\text{AAQIB}$ (Distinct letters $= \{A, Q, I, B\}$, $n = 4$, $2^4 = 16$):
$$P(\text{AAQIB}) = \mathbf{\{\emptyset, \{A\}, \{Q\}, \{I\}, \{B\}, \{A, Q\}, \{A, I\}, \{A, B\}, \{Q, I\}, \{Q, B\}, \{I, B\}, \{A, Q, I\}, \{A, Q, B\}, \{A, I, B\}, \{Q, I, B\}, \{A, Q, I, B\}\}}$$
Q6. Find the power set of $\{0, \{1\}\}$.
Here set has 2 elements: $0$ and the set $\{1\}$. Number of subsets $= 2^2 = 4$.
$$P(\{0, \{1\}\}) = \mathbf{\{\emptyset, \{0\}, \{\{1\}\}, \{0, \{1\}\}\}}$$
$$P(\{0, \{1\}\}) = \mathbf{\{\emptyset, \{0\}, \{\{1\}\}, \{0, \{1\}\}\}}$$
Q7. A is the set of first 15 letters of English alphabet, and B is the set of consonants in A:
(i) List elements of A and B:
- $A = \mathbf{\{a, b, c, d, e, f, g, h, i, j, k, l, m, n, o\}}$ ($n(A) = 15$)
- Vowels in $A = \{a, e, i, o\}$ $\implies B = \mathbf{\{b, c, d, f, g, h, j, k, l, m, n\}}$ ($n(B) = 11$)
(ii) Number of elements in $P(A)$ and $P(B)$:
- $n(P(A)) = 2^{15} = \mathbf{32,768}$
- $n(P(B)) = 2^{11} = \mathbf{2,048}$
- $A = \mathbf{\{a, b, c, d, e, f, g, h, i, j, k, l, m, n, o\}}$ ($n(A) = 15$)
- Vowels in $A = \{a, e, i, o\}$ $\implies B = \mathbf{\{b, c, d, f, g, h, j, k, l, m, n\}}$ ($n(B) = 11$)
(ii) Number of elements in $P(A)$ and $P(B)$:
- $n(P(A)) = 2^{15} = \mathbf{32,768}$
- $n(P(B)) = 2^{11} = \mathbf{2,048}$
Exercise 5.2 • Verification of Set Laws & De Morgan’s Laws
Q1 & Q2. Commutative Properties of Union ($A \cup B = B \cup A$) and Intersection ($A \cap B = B \cap A$):
For sets $A = \{1, 2, 3, 4, 5\}, B = \{2, 4, 6, 8\}$:
- $A \cup B = \{1, 2, 3, 4, 5, 6, 8\} = B \cup A$ (Verified ✓)
- $A \cap B = \{2, 4\} = B \cap A$ (Verified ✓)
- $A \cup B = \{1, 2, 3, 4, 5, 6, 8\} = B \cup A$ (Verified ✓)
- $A \cap B = \{2, 4\} = B \cap A$ (Verified ✓)
Q3 & Q4. Associative Properties of Union $(A \cup B) \cup C = A \cup (B \cup C)$ and Intersection $(A \cap B) \cap C = A \cap (B \cap C)$:
For $A = \{a, b, c\}, B = \{b, c, d\}, C = \{c, d, e\}$:
- $A \cup B = \{a, b, c, d\} \implies (A \cup B) \cup C = \{a, b, c, d, e\}$
- $B \cup C = \{b, c, d, e\} \implies A \cup (B \cup C) = \{a, b, c, d, e\}$ $\implies$ LHS = RHS ✓
- $A \cap B = \{b, c\} \implies (A \cap B) \cap C = \{c\}$
- $B \cap C = \{c, d\} \implies A \cap (B \cap C) = \{c\}$ $\implies$ LHS = RHS ✓
- $A \cup B = \{a, b, c, d\} \implies (A \cup B) \cup C = \{a, b, c, d, e\}$
- $B \cup C = \{b, c, d, e\} \implies A \cup (B \cup C) = \{a, b, c, d, e\}$ $\implies$ LHS = RHS ✓
- $A \cap B = \{b, c\} \implies (A \cap B) \cap C = \{c\}$
- $B \cap C = \{c, d\} \implies A \cap (B \cap C) = \{c\}$ $\implies$ LHS = RHS ✓
Q5 & Q6. Distributive Laws of Union over Intersection and Intersection over Union:
(1) $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$:
Let $A = \{1, 3, 5\}, B = \{2, 3, 4\}, C = \{3, 4, 5\}$.
- $B \cap C = \{3, 4\} \implies A \cup (B \cap C) = \{1, 3, 4, 5\}$
- $A \cup B = \{1, 2, 3, 4, 5\}$, $A \cup C = \{1, 3, 4, 5\} \implies (A \cup B) \cap (A \cup C) = \{1, 3, 4, 5\}$ $\implies$ LHS = RHS ✓
(2) $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$:
- $B \cup C = \{2, 3, 4, 5\} \implies A \cap (B \cup C) = \{3, 5\}$
- $A \cap B = \{3\}$, $A \cap C = \{3, 5\} \implies (A \cap B) \cup (A \cap C) = \{3, 5\}$ $\implies$ LHS = RHS ✓
Let $A = \{1, 3, 5\}, B = \{2, 3, 4\}, C = \{3, 4, 5\}$.
- $B \cap C = \{3, 4\} \implies A \cup (B \cap C) = \{1, 3, 4, 5\}$
- $A \cup B = \{1, 2, 3, 4, 5\}$, $A \cup C = \{1, 3, 4, 5\} \implies (A \cup B) \cap (A \cup C) = \{1, 3, 4, 5\}$ $\implies$ LHS = RHS ✓
(2) $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$:
- $B \cup C = \{2, 3, 4, 5\} \implies A \cap (B \cup C) = \{3, 5\}$
- $A \cap B = \{3\}$, $A \cap C = \{3, 5\} \implies (A \cap B) \cup (A \cap C) = \{3, 5\}$ $\implies$ LHS = RHS ✓
Q7, Q8, Q9. Verification of De Morgan's Laws ($(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$):
Let $U = \{1, 2, 3, \dots, 10\}, A = \{2, 4, 6, 8, 10\}, B = \{1, 2, 3, 4, 5\}$.
Law 1: $(A \cup B)' = A' \cap B'$
- $A \cup B = \{1, 2, 3, 4, 5, 6, 8, 10\} \implies \text{LHS} = (A \cup B)' = \{7, 9\}$
- $A' = U - A = \{1, 3, 5, 7, 9\}$
- $B' = U - B = \{6, 7, 8, 9, 10\}$
- $\text{RHS} = A' \cap B' = \{7, 9\} \implies$ LHS = RHS (Verified ✓)
Law 2: $(A \cap B)' = A' \cup B'$
- $A \cap B = \{2, 4\} \implies \text{LHS} = (A \cap B)' = \{1, 3, 5, 6, 7, 8, 9, 10\}$
- $\text{RHS} = A' \cup B' = \{1, 3, 5, 7, 9\} \cup \{6, 7, 8, 9, 10\} = \{1, 3, 5, 6, 7, 8, 9, 10\} \implies$ LHS = RHS (Verified ✓)
Law 1: $(A \cup B)' = A' \cap B'$
- $A \cup B = \{1, 2, 3, 4, 5, 6, 8, 10\} \implies \text{LHS} = (A \cup B)' = \{7, 9\}$
- $A' = U - A = \{1, 3, 5, 7, 9\}$
- $B' = U - B = \{6, 7, 8, 9, 10\}$
- $\text{RHS} = A' \cap B' = \{7, 9\} \implies$ LHS = RHS (Verified ✓)
Law 2: $(A \cap B)' = A' \cup B'$
- $A \cap B = \{2, 4\} \implies \text{LHS} = (A \cap B)' = \{1, 3, 5, 6, 7, 8, 9, 10\}$
- $\text{RHS} = A' \cup B' = \{1, 3, 5, 7, 9\} \cup \{6, 7, 8, 9, 10\} = \{1, 3, 5, 6, 7, 8, 9, 10\} \implies$ LHS = RHS (Verified ✓)
Q10. Find $A - B$ and $B - A$:
(i) $A = \{1, 2, 3, 4, 5\}, B = \{2, 4, 6, 8\}$:
- $A - B = \mathbf{\{1, 3, 5\}}$
- $B - A = \mathbf{\{6, 8\}}$
(ii) $A = \{a, b, c, d\}, B = \{c, d, e, f\}$:
- $A - B = \mathbf{\{a, b\}}$
- $B - A = \mathbf{\{e, f\}}$
- $A - B = \mathbf{\{1, 3, 5\}}$
- $B - A = \mathbf{\{6, 8\}}$
(ii) $A = \{a, b, c, d\}, B = \{c, d, e, f\}$:
- $A - B = \mathbf{\{a, b\}}$
- $B - A = \mathbf{\{e, f\}}$
Exercise 5.3 • Venn Diagrams & Shading Analysis
Q1. (a) Find $A \cup B$ and $A \cap B$. (b) Represent through Venn diagrams:
(i) $A = \{0, 1, 2, 3, 4, 5, 6\}, B = \{4, 5, 6, 7, 8\}$: (Overlapping Sets)
- $A \cup B = \mathbf{\{0, 1, 2, 3, 4, 5, 6, 7, 8\}}$ (Venn Diagram: Entire area of both circles $A$ and $B$ is shaded)
- $A \cap B = \mathbf{\{4, 5, 6\}}$ (Venn Diagram: Only the overlapping common intersection region is shaded)
(ii) $A = \{a, b, c, d\}, B = \{e, f, g\}$: (Disjoint Sets)
- $A \cup B = \mathbf{\{a, b, c, d, e, f, g\}}$ (Venn Diagram: Both separated circles are shaded)
- $A \cap B = \mathbf{\emptyset}$ (Venn Diagram: No shading in either circle)
(iii) $A = \{1, 2, \dots, 8\}, B = \{0, 1, 2, \dots, 9\}$: (Subset $A \subset B$)
- $A \cup B = B = \mathbf{\{0, 1, 2, \dots, 9\}}$ (Venn Diagram: Outer circle $B$ including inner circle $A$ is shaded)
- $A \cap B = A = \mathbf{\{1, 2, \dots, 8\}}$ (Venn Diagram: Only inner circle $A$ is shaded)
- $A \cup B = \mathbf{\{0, 1, 2, 3, 4, 5, 6, 7, 8\}}$ (Venn Diagram: Entire area of both circles $A$ and $B$ is shaded)
- $A \cap B = \mathbf{\{4, 5, 6\}}$ (Venn Diagram: Only the overlapping common intersection region is shaded)
(ii) $A = \{a, b, c, d\}, B = \{e, f, g\}$: (Disjoint Sets)
- $A \cup B = \mathbf{\{a, b, c, d, e, f, g\}}$ (Venn Diagram: Both separated circles are shaded)
- $A \cap B = \mathbf{\emptyset}$ (Venn Diagram: No shading in either circle)
(iii) $A = \{1, 2, \dots, 8\}, B = \{0, 1, 2, \dots, 9\}$: (Subset $A \subset B$)
- $A \cup B = B = \mathbf{\{0, 1, 2, \dots, 9\}}$ (Venn Diagram: Outer circle $B$ including inner circle $A$ is shaded)
- $A \cap B = A = \mathbf{\{1, 2, \dots, 8\}}$ (Venn Diagram: Only inner circle $A$ is shaded)
Q2. Find $C - D$ and $D - C$ and represent through Venn Diagrams:
(i) $C = \{11, 12, 13, 14, 15\}, D = \{14, 16, 18, 20\}$:
- $C - D = \mathbf{\{11, 12, 13, 15\}}$ (Shade circle $C$ excluding intersection with $D$)
- $D - C = \mathbf{\{16, 18, 20\}}$ (Shade circle $D$ excluding intersection with $C$)
(ii) $C = \{-1, -2, -3\}, D = \{0, -1, -2, -3, -4, -5\}$: ($C \subset D$)
- $C - D = \mathbf{\emptyset}$ (No shading)
- $D - C = \mathbf{\{0, -4, -5\}}$ (Shade outer ring of $D$ outside inner circle $C$)
- $C - D = \mathbf{\{11, 12, 13, 15\}}$ (Shade circle $C$ excluding intersection with $D$)
- $D - C = \mathbf{\{16, 18, 20\}}$ (Shade circle $D$ excluding intersection with $C$)
(ii) $C = \{-1, -2, -3\}, D = \{0, -1, -2, -3, -4, -5\}$: ($C \subset D$)
- $C - D = \mathbf{\emptyset}$ (No shading)
- $D - C = \mathbf{\{0, -4, -5\}}$ (Shade outer ring of $D$ outside inner circle $C$)
Q3 & Q4. Represent Complements and De Morgan’s Laws through Venn Diagrams:
- **$X'$:** Entire rectangle $U$ outside circle $X$ is shaded.
- **$(X \cup Y)'$:** The entire rectangle $U$ outside both circles $X$ and $Y$ is shaded.
- **$(X \cap Y)'$:** The entire universe $U$ is shaded except for the small overlapping lens $X \cap Y$.
- **$(X - Y)'$:** Everything in $U$ except the crescent $X - Y$ is shaded.
- **$(X \cup Y)'$:** The entire rectangle $U$ outside both circles $X$ and $Y$ is shaded.
- **$(X \cap Y)'$:** The entire universe $U$ is shaded except for the small overlapping lens $X \cap Y$.
- **$(X - Y)'$:** Everything in $U$ except the crescent $X - Y$ is shaded.
Q5. Venn Diagram Shading Operations Guide:
- (i) $A'$: Shade all of $U$ outside circle $A$.
- (ii) $A - B$: Shade circle $A$ excluding intersection with $B$.
- (iii) $C \cap D$: Shade only the overlapping region between $C$ and $D$.
- (iv) $X \cup Y$ ($X \subseteq Y$): Shade circle $Y$ entirely.
- (v) $(L - M)'$ ($L, M$ disjoint): Since $L - M = L$, shade everything in $U$ outside circle $L$.
- (vi) $(E \cup F)'$: Shade the background of $U$ outside both $E$ and $F$.
- (ii) $A - B$: Shade circle $A$ excluding intersection with $B$.
- (iii) $C \cap D$: Shade only the overlapping region between $C$ and $D$.
- (iv) $X \cup Y$ ($X \subseteq Y$): Shade circle $Y$ entirely.
- (v) $(L - M)'$ ($L, M$ disjoint): Since $L - M = L$, shade everything in $U$ outside circle $L$.
- (vi) $(E \cup F)'$: Shade the background of $U$ outside both $E$ and $F$.
Exercise 5.4 • Real-World Applications of Sets
Q1. School employees transport survey ($W \subset C \subset U$): $n(W) = 26$, $n(C - W) = 46$, public transport outside $= 28$.
- (a) Total employees in school: $n(U) = 26 + 46 + 28 = \mathbf{100\text{ employees}}$.
- (b) Employees having motor cars: $n(C) = 26 + 46 = \mathbf{72\text{ employees}}$.
- (c) Employees having white motor cars: $n(W) = \mathbf{26\text{ employees}}$.
- (d) Motor cars with color other than white: $n(C - W) = \mathbf{46\text{ employees}}$.
- (b) Employees having motor cars: $n(C) = 26 + 46 = \mathbf{72\text{ employees}}$.
- (c) Employees having white motor cars: $n(W) = \mathbf{26\text{ employees}}$.
- (d) Motor cars with color other than white: $n(C - W) = \mathbf{46\text{ employees}}$.
Q2. Drawing ($D$) and Agriculture ($A$) survey: Only $D = 17$, $D \cap A = 5$, Only $A = 18$.
- (a) Total students in the class: $17 + 5 + 18 = \mathbf{40\text{ students}}$.
- (b) Students studying both subjects: $n(D \cap A) = \mathbf{5\text{ students}}$.
- (c) Students studying at least one subject: $n(D \cup A) = 17 + 5 + 18 = \mathbf{40\text{ students}}$.
- (d) Students studying only agriculture: $n(A - D) = \mathbf{18\text{ students}}$.
- (e) Students who do not study drawing: $40 - (17 + 5) = \mathbf{18\text{ students}}$.
- (b) Students studying both subjects: $n(D \cap A) = \mathbf{5\text{ students}}$.
- (c) Students studying at least one subject: $n(D \cup A) = 17 + 5 + 18 = \mathbf{40\text{ students}}$.
- (d) Students studying only agriculture: $n(A - D) = \mathbf{18\text{ students}}$.
- (e) Students who do not study drawing: $40 - (17 + 5) = \mathbf{18\text{ students}}$.
Q3. People at interview: English ($A$) only $= 20$, Both ($A \cap B$) $= 10$, Urdu ($B$) only $= 30$, Neither $= 15$.
- (a) Total people at interview: $n(U) = 20 + 10 + 30 + 15 = \mathbf{75\text{ people}}$.
- (b) Language comprehension breakdown:
• (i) Understand English: $20 + 10 = \mathbf{30\text{ people}}$
• (ii) Understand Urdu: $30 + 10 = \mathbf{40\text{ people}}$
• (iii) Both English and Urdu: $\mathbf{10\text{ people}}$
• (iv) English but not Urdu: $\mathbf{20\text{ people}}$
• (v) Neither English nor Urdu: $\mathbf{15\text{ people}}$
• (vi) English or Urdu ($A \cup B$): $20 + 10 + 30 = \mathbf{60\text{ people}}$
- (b) Language comprehension breakdown:
• (i) Understand English: $20 + 10 = \mathbf{30\text{ people}}$
• (ii) Understand Urdu: $30 + 10 = \mathbf{40\text{ people}}$
• (iii) Both English and Urdu: $\mathbf{10\text{ people}}$
• (iv) English but not Urdu: $\mathbf{20\text{ people}}$
• (v) Neither English nor Urdu: $\mathbf{15\text{ people}}$
• (vi) English or Urdu ($A \cup B$): $20 + 10 + 30 = \mathbf{60\text{ people}}$
Q4. Fruit juice choices: Orange ($A$) only $= 5$, Both ($A \cap B$) $= 4$, Carrot ($B$) only $= 8$, Neither $= 3$.
- (a) Total students: $5 + 4 + 8 + 3 = \mathbf{20\text{ students}}$.
- (b) Breakdown:
• (i) Orange juice: $5 + 4 = \mathbf{9\text{ students}}$
• (ii) Carrot juice: $8 + 4 = \mathbf{12\text{ students}}$
• (iii) Orange and carrot juice: $\mathbf{4\text{ students}}$
• (iv) Orange or carrot juice: $5 + 4 + 8 = \mathbf{17\text{ students}}$
• (v) Neither orange nor carrot: $\mathbf{3\text{ students}}$
• (vi) Orange but not carrot: $\mathbf{5\text{ students}}$
• (vii) Only orange juice: $\mathbf{5\text{ students}}$
• (viii) Only carrot juice: $\mathbf{8\text{ students}}$
- (b) Breakdown:
• (i) Orange juice: $5 + 4 = \mathbf{9\text{ students}}$
• (ii) Carrot juice: $8 + 4 = \mathbf{12\text{ students}}$
• (iii) Orange and carrot juice: $\mathbf{4\text{ students}}$
• (iv) Orange or carrot juice: $5 + 4 + 8 = \mathbf{17\text{ students}}$
• (v) Neither orange nor carrot: $\mathbf{3\text{ students}}$
• (vi) Orange but not carrot: $\mathbf{5\text{ students}}$
• (vii) Only orange juice: $\mathbf{5\text{ students}}$
• (viii) Only carrot juice: $\mathbf{8\text{ students}}$
Review Exercise 5 • Comprehensive Mastery Assessment
Q1. Encircle the correct answer:
(i) $\{0, 1, 2, 3, 4, 5\}$ is in ________ form → (c) tabular
(ii) Tabular form of $\{x : x \in \mathbb{N} \land x \le 3\}$ → (d) $\{1, 2, 3\}$
(iii) $\{x : x \in \mathbb{E} \land 4 < x \le 6\} =$ → (b) $\{6\}$
(iv) In descriptive form, $\{1, 2, 3, 6, 9, 18\} =$ → (c) set of factors of 18
(v) $\{x : x \in A \land x \notin B\} =$ → (c) $A - B$
(vi) Which is NOT true in general? → (c) $A - B = B - A$
(vii) For $A = \{1, 2, 3, 4, 5\}, B = \{1, 2, 5, 9\}$, sets are → (d) overlapping
(viii) If $A = \{1, \dots, 10\}, B = \{1, 3, 5, 7, 9\}$, then $A - B =$ → (a) $\{2, 4, 6, 8, 10\}$
(ix) $\emptyset - \{a, b, c\} =$ → (d) $\emptyset$
(x) If $A$ and $B$ are disjoint sets, then → (d) $A \cap B = \emptyset$
(xi) If $U$ is universal set, $X' =$ → (b) $U - X$
(xii) $D \cup D' =$ → (b) $U$
(xiii) $A \cup A =$ → (b) $A$
(xiv) Relation between $A$ and $B$ in overlapping diagram → (a) overlapping
(xv) Shaded part excluding $B$ → (a) $A - B$
(xvi) Ring of subset circle → (b) $Q - P$
(xvii) Outside both disjoint circles → (c) $(A \cup B)'$
(ii) Tabular form of $\{x : x \in \mathbb{N} \land x \le 3\}$ → (d) $\{1, 2, 3\}$
(iii) $\{x : x \in \mathbb{E} \land 4 < x \le 6\} =$ → (b) $\{6\}$
(iv) In descriptive form, $\{1, 2, 3, 6, 9, 18\} =$ → (c) set of factors of 18
(v) $\{x : x \in A \land x \notin B\} =$ → (c) $A - B$
(vi) Which is NOT true in general? → (c) $A - B = B - A$
(vii) For $A = \{1, 2, 3, 4, 5\}, B = \{1, 2, 5, 9\}$, sets are → (d) overlapping
(viii) If $A = \{1, \dots, 10\}, B = \{1, 3, 5, 7, 9\}$, then $A - B =$ → (a) $\{2, 4, 6, 8, 10\}$
(ix) $\emptyset - \{a, b, c\} =$ → (d) $\emptyset$
(x) If $A$ and $B$ are disjoint sets, then → (d) $A \cap B = \emptyset$
(xi) If $U$ is universal set, $X' =$ → (b) $U - X$
(xii) $D \cup D' =$ → (b) $U$
(xiii) $A \cup A =$ → (b) $A$
(xiv) Relation between $A$ and $B$ in overlapping diagram → (a) overlapping
(xv) Shaded part excluding $B$ → (a) $A - B$
(xvi) Ring of subset circle → (b) $Q - P$
(xvii) Outside both disjoint circles → (c) $(A \cup B)'$
Q2. If $U = \{1, \dots, 10\}, A = \{1, 2, 3, 4, 5\}, B = \{4, 5, 6, 7, 8\}$, evaluate:
- (i) $A \cup B = \mathbf{\{1, 2, 3, 4, 5, 6, 7, 8\}}$
- (ii) $A \cap B = \mathbf{\{4, 5\}}$
- (iii) $A - B = \mathbf{\{1, 2, 3\}}$
- (iv) $B - A = \mathbf{\{6, 7, 8\}}$
- (v) $A' = U - A = \mathbf{\{6, 7, 8, 9, 10\}}$
- (vi) $B' = U - B = \mathbf{\{1, 2, 3, 9, 10\}}$
- (vii) $A' \cap B' = \mathbf{\{9, 10\}}$
- (viii) $(A - B)' = U - \{1, 2, 3\} = \mathbf{\{4, 5, 6, 7, 8, 9, 10\}}$
- (ix) $A' \cup B' = \mathbf{\{1, 2, 3, 6, 7, 8, 9, 10\}}$
- (x) $(B - A)' = U - \{6, 7, 8\} = \mathbf{\{1, 2, 3, 4, 5, 9, 10\}}$
- (xi) $(A \cup B)' = \mathbf{\{9, 10\}}$
- (xii) $(A \cap B)' = \mathbf{\{1, 2, 3, 6, 7, 8, 9, 10\}}$
- (ii) $A \cap B = \mathbf{\{4, 5\}}$
- (iii) $A - B = \mathbf{\{1, 2, 3\}}$
- (iv) $B - A = \mathbf{\{6, 7, 8\}}$
- (v) $A' = U - A = \mathbf{\{6, 7, 8, 9, 10\}}$
- (vi) $B' = U - B = \mathbf{\{1, 2, 3, 9, 10\}}$
- (vii) $A' \cap B' = \mathbf{\{9, 10\}}$
- (viii) $(A - B)' = U - \{1, 2, 3\} = \mathbf{\{4, 5, 6, 7, 8, 9, 10\}}$
- (ix) $A' \cup B' = \mathbf{\{1, 2, 3, 6, 7, 8, 9, 10\}}$
- (x) $(B - A)' = U - \{6, 7, 8\} = \mathbf{\{1, 2, 3, 4, 5, 9, 10\}}$
- (xi) $(A \cup B)' = \mathbf{\{9, 10\}}$
- (xii) $(A \cap B)' = \mathbf{\{1, 2, 3, 6, 7, 8, 9, 10\}}$
Q3. For $U = \{1, \dots, 10\}, X = \{1, 2, 3, 4, 5, 6\}, Y = \{4, 5, 6, 7, 8, 9, 10\}$, prove:
- (i) $X - Y = X \cap Y'$: $X - Y = \{1, 2, 3\}$. $Y' = \{1, 2, 3\} \implies X \cap Y' = \{1, 2, 3\}$ (Proved ✓)
- (ii) $Y - X = Y \cap X'$: $Y - X = \{7, 8, 9, 10\}$. $X' = \{7, 8, 9, 10\} \implies Y \cap X' = \{7, 8, 9, 10\}$ (Proved ✓)
- (iii) $(X \cup Y)' = X' \cap Y'$: $X \cup Y = U \implies (X \cup Y)' = \emptyset$. $X' \cap Y' = \{7, 8, 9, 10\} \cap \{1, 2, 3\} = \emptyset$ (Proved ✓)
- (iv) $X - Y' = Y - X'$: $X - Y' = \{1, \dots, 6\} - \{1, 2, 3\} = \{4, 5, 6\}$. $Y - X' = \{4, \dots, 10\} - \{7, \dots, 10\} = \{4, 5, 6\}$ (Proved ✓)
- (v) $(X')' = X$: $X' = \{7, 8, 9, 10\} \implies (X')' = U - \{7, 8, 9, 10\} = \{1, 2, 3, 4, 5, 6\} = X$ (Proved ✓)
- (ii) $Y - X = Y \cap X'$: $Y - X = \{7, 8, 9, 10\}$. $X' = \{7, 8, 9, 10\} \implies Y \cap X' = \{7, 8, 9, 10\}$ (Proved ✓)
- (iii) $(X \cup Y)' = X' \cap Y'$: $X \cup Y = U \implies (X \cup Y)' = \emptyset$. $X' \cap Y' = \{7, 8, 9, 10\} \cap \{1, 2, 3\} = \emptyset$ (Proved ✓)
- (iv) $X - Y' = Y - X'$: $X - Y' = \{1, \dots, 6\} - \{1, 2, 3\} = \{4, 5, 6\}$. $Y - X' = \{4, \dots, 10\} - \{7, \dots, 10\} = \{4, 5, 6\}$ (Proved ✓)
- (v) $(X')' = X$: $X' = \{7, 8, 9, 10\} \implies (X')' = U - \{7, 8, 9, 10\} = \{1, 2, 3, 4, 5, 6\} = X$ (Proved ✓)
Q4. Given $U = \{0, 1, \dots, 11\}$, $P = \text{multiples of 3} < 15$, $Q = \text{even numbers} < 10$:
- $P = \{0, 3, 6, 9\}$ (in $U$), $Q = \{0, 2, 4, 6, 8\}$
- (a) Multiples of 3: $\mathbf{\{0, 3, 6, 9\}}$
- (b) Even numbers: $\mathbf{\{0, 2, 4, 6, 8\}}$
- (c) Both multiple of 3 and even ($P \cap Q$): $\mathbf{\{0, 6\}}$
- (d) Neither even nor multiple of 3 ($(P \cup Q)'$): $P \cup Q = \{0, 2, 3, 4, 6, 8, 9\} \implies \mathbf{\{1, 5, 7, 10, 11\}}$
- (a) Multiples of 3: $\mathbf{\{0, 3, 6, 9\}}$
- (b) Even numbers: $\mathbf{\{0, 2, 4, 6, 8\}}$
- (c) Both multiple of 3 and even ($P \cap Q$): $\mathbf{\{0, 6\}}$
- (d) Neither even nor multiple of 3 ($(P \cup Q)'$): $P \cup Q = \{0, 2, 3, 4, 6, 8, 9\} \implies \mathbf{\{1, 5, 7, 10, 11\}}$
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