Mastering Set Theory: Study the Notes & Reinforce with Nairafame AI.

SETS

Definition: A Set can be defined as a group or collection of well-defined objects or numbers.

Example 1

1. A collection of books in a public library
2. The set of all Natural numbers:
   $$S = N = \{1,2,3,...\}$$
   Swipe left/right to see full formula →
3. The set of integers denoted by:
   $$\mu = \{...,-4,-3,-2,-1,0,1,2,3,...\}$$
4. The set of odd integers:
   $$\mu = \{...,-7,-5,-3,-1,0,1,3,5,7,...\}$$
5. The set of all vowels in the English Alphabet:
   $$V = \{a,e,i,o,u\}$$
6. The set of all the colors of the rainbow:
   $$C = \{Red, Orange, Yellow, Green, Blue, Indigo, Violet\}$$
7. Group of girls in Varce Girl Learning Hub.
8. The set of days in a week:
   $$D = \{\text{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}\}$$
9. The set of planets in our Solar System:
   $$P = \{\text{Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}\}$$

10. The set of prime numbers less than 20:
    $$P_{<20 11="" 13="" 17="" 19="" 3="" 5="" 7="" div="">
11. The set of distinct letters in the word "MATHEMATICS":
    $$L = \{M, A, T, H, E, I, C, S\}$$
    Note: In sets, we usually don't repeat elements, so even though 'M', 'A', and 'T' appear twice in the word, they only appear once in the set!
12. The set of denominations of US circulating coins:
    $$C = \{\text{1¢, 5¢, 10¢, 25¢, 50¢, \$1}\}$$

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Describing a Set

1. Roster Form

This method involves listing every element or member within curly brackets \(\{\}\).

• (a) \(F = \{\text{Mango, Grape, Orange, Guava, Apple}\}\)
• (b) \(M = \{6, 9, 12, 15, 18\}\)

2. Statement Form

This is when sets are described using words.

• (a) \(M = \{\text{Multiples of 3 from 3 to 18}\}\)
• (b) \(C = \{\text{Consonants}\}\)
• (c) \(D = \{\text{Domestic Animals}\}\)
• (d) \(A = \{\text{Set of Odd numbers less than 9}\}\)

3. Set Builder Form

$$A = \{x : x = 2n, \text{ } n \in N \text{ and } 1 \leq n \leq 4\}$$

Calculations:

• If \(n = 1\), then \(x = 2(1) = 2\)
• If \(n = 2\), then \(x = 2(2) = 4\)
• If \(n = 3\), then \(x = 2(3) = 6\)
• If \(n = 4\), then \(x = 2(4) = 8\)

Resulting Roster Form: \(A = \{2, 4, 6, 8\}\)

Example:

\(B = \{x: -10 < x \leq 3, x \text{ is an integer}\}\)

Roster Form:

\(B = \{-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3\}\)

Note: \(-10\) is not included because the symbol is \(<\), but \(3\) is included because the symbol is \(\leq\).

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TYPES OF SETS

(i) Finite Sets

A set that has a countable number of members.

• (a) The English Alphabet: \(S = \{a, b, c, d, ..., z\}\)
• (b) English Vowels: \(V = \{a, e, i, o, u\}\)
• (c) Factors of 6: \(A = \{1, 2, 3, 6\}\)

(ii) Infinite Sets

A set whose elements cannot be counted. We use an ellipsis (...) to show it continues forever.

• (a) Whole Numbers: \(W = \{0, 1, 2, 3, 4, ...\}\)
• (b) Odd numbers greater than 5: \(B = \{7, 9, 11, 13, 15, ...\}\)

(iii) Singleton Set

A set that contains exactly one element.

Example: Vice-Presidents in Nigeria (Current).

(iv) Empty Set (or Null Set)

A set is Empty or Null if it contains no elements.

Symbols: { } or \(\emptyset\)

Note: Do not put the symbol inside the brackets, like \(\{\emptyset\}\)!

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Membership of a Set

The symbol ∈ means "member of".

If \(A = \{1, 4, 9, 16\}\), then \(1 \in A\) and \(4 \in A\).

The symbol ∉ means "not a member of".

Example: \(5 \notin A\).

SETS & VENN DIAGRAMS

Venn Diagrams

A Venn Diagram helps us visualize relationships between sets.

1. Union of Three Sets

$$A \cup B \cup C$$

All regions inside A, B, and C are shaded.

2. Intersection of Three Sets

$$A \cap B \cap C$$

Only the middle overlapping part is shaded.

Example 6

Draw a Venn Diagram to show the set A = {1, 3, 5, 7, 9}

 

Subset of a Set

A set \(T\) is a subset of \(S\) if every element of \(T\) is in \(S\).

$$T \subset S$$

Example:

\(S = \{a, b, c, ..., z\}\)

\(V = \{a, e, i, o, u\}\)

Therefore, \(V \subset S\)

Universal Set

The universal set contains all elements under consideration.

$$\mu = \{1,2,3,...,10\}$$

Complement of a Set

Elements not in set A.

$$A' = \mu - A$$

Example:

\(\mu = \{1,2,3,4,5,6,7,8\}\)

\(A = \{1,3,5\}\)

$$A' = \{2,4,6,7,8\}$$

Number of Elements (Cardinality)

$$n(A)$$

Example: \(A = \{a,e,i,o,u\}\)

\(n(A) = 5\)

Disjoint Sets

Sets with no common elements.

$$A \cap B = \emptyset$$

Equal and Equivalent Sets

Equal Sets

Same elements.

$$A = B$$

Equivalent Sets

Same number of elements.

$$n(A) = n(B)$$

All equal sets are equivalent, but not all equivalent sets are equal.

Example: Days of the Week

\(U = \{\text{Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}\}\)

\(B = \{\text{Sunday, Saturday}\}\)

\(B' = \{\text{Monday, Tuesday, Wednesday, Thursday, Friday}\}\)

$$n(B) + n(B') = n(U)$$

$$2 + 5 = 7$$

Months Example

\(A = \{\text{January, June, July}\}\)

\(B = \{\text{September, October, November, December}\}\)

\(n(A) = 3,\quad n(B') = 8\)

$$n(A) + n(B') = 11$$

EXAMPLE 15: ECOWAS SUB-REGION

Given that:

• \(U = \{\text{all countries in the ECOWAS sub-region}\}\)
• \(X = \{\text{all Anglophone countries in West Africa}\}\)
• \(Y = \{\text{all Francophone countries in West Africa}\}\)

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Questions & Solutions

i. List all elements in \(U\):

\(U = \{\text{Benin, Burkina Faso, Cape Verde, Cote d'Ivoire, The Gambia, Ghana, Guinea, Guinea-Bissau, Liberia, Mali, Niger, Nigeria, Senegal, Sierra Leone, Togo}\}\)

ii. List all elements in \(X\):

These are the English-speaking (Anglophone) countries in the region.

\(X = \{\text{The Gambia, Ghana, Liberia, Nigeria, Sierra Leone}\}\)

UNION (∪) OF SETS

Let A and B be any two sets; the Union of A and B written \(A \cup B\) is the set of elements in either A or B.

Figure (i): Shaded region representing \(A \cup B\)

NB: The Union of set A and B is the set of all members that belong to A and B or to both A and B.

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INTERSECTION (∩) OF SETS

The Intersection of set A and B is the set of members that are common to both A and B.

Let A and B be any two sets, the intersection of A and B written \(A \cap B\) is the sets of element in both A and B.

 

Figure (ii): Shaded region representing \(A \cap B\)

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Relationship Between Union and Intersection

Let \(A = \{x, y\}\) be a set, and \(B = \{y, z\}\) another set.

 

\(A \cup B = \{x, y, z\}\)

\(A \cap B = \{y\}\)

The General Formula:

\(n(A \cup B) = n(A) + n(B) - n(A \cap B)\)

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Example 16:

In a class of 80 students, every student had to study Economics and Geography or both Economics and Geography. If 65 students studied Economics and 50 students Geography, how many studied both subjects?

Solution;

Let E represent Economics, G represent Geography

\(n(E \cup G) = 80\)
\(n(E) = 65\)
\(n(G) = 50\)
Let \(x\) represent students who study both, i.e., \(n(E \cap G) = x\)

Method 1: Using the General Formula

\[n(E \cup G) = n(E) + n(G) - n(E \cap G)\] \(80 = 65 + 50 - x\)
\(80 = 115 - x\)
\(x = 115 - 80\)
\(x = 35\)

### OR ###

Method 2: Using the Venn Diagram Approach

(Summing the individual parts: Economics only + Both + Geography only)

\(80 = (65 - x) + x + (50 - x)\)
\(80 = 65 + 50 - x\)
\(80 = 115 - x\)
\(x = 115 - 80\)
\(x = 35\)

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Example 17;

A survey of 100 students in an institution shows that 80 students speak Hausa and 20 students speak Igbo while 9 students speak both languages. How many students speak neither Hausa nor Igbo?

Solution

Let H represent Hausa and I represent Igbo.

• \(n(H \cap I) = 9\) (Those who speak both)
• \(n(H \text{ only}) = 80 - 9 = 71\)
• \(n(I \text{ only}) = 20 - 9 = 11\)
• \(n(U) = 100\) (Total students in the survey)

Calculating the number of students who speak neither:

\(n(\text{Neither H nor I}) = 100 - 71 - 11 - 9\)
\(n(\text{Neither H nor I}) = 9\)

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Example 18;

a) Show that \(A = \{x : -4 \leq x \leq 5\}\) is a subset of \(B = \{x : -7 \leq x \leq 6\}\)
b) Find \(A \cap B\)

Solution

Listing the elements of the sets:

\(A = \{-4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}\)
\(B = \{-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6\}\)

a) \(A \subset B = ?\)

Since all elements of \(A\) are in \(B\), then \(A \subset B\).

b) Find \(A \cap B\)

\(A \cap B = \{-4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}\)

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Example 19;

From a sample of 78 young people who listened to Music, 42 listened to Cassette player, 42 to records and 40 to live music. 8 students listened to cassettes only, 10 to records only, 20 to records and Cassettes only, 8 to music from all the three sources. Find the number of students who listened to live music only.

Solution;

Using Venn Diagram Method:

Let \(n(C)\) = Cassettes player (42)
Let \(n(R)\) = Records (42)
Let \(n(L)\) = Live music (40)

The Venn Diagram Components:

• \(U = 78\)
• \(C \text{ only} = 8\)
• \(R \text{ only} = 10\)
• \(n(C \cap R \text{ only}) = 20\)
• \(n(C \cap R \cap L) = 8\)
• Let \(x\) = live music only
• Let \(y = n(C \cap L \text{ only})\) and \(z = n(R \cap L \text{ only})\)

 

Calculations:

To find \(x\), we first find \(y\) and \(z\) from the totals of circles \(C\) and \(R\):

For Circle C:
\(8 + 20 + 8 + y = 42\)
\(36 + y = 42 \implies \mathbf{y = 6}\)

For Circle R:
\(10 + 20 + 8 + z = 42\)
\(38 + z = 42 \implies \mathbf{z = 4}\)

Now, solving for \(x\) in Circle L:

\(x + y + z + 8 = 40\)
\(x + 6 + 4 + 8 = 40\)
\(x + 18 = 40\)
\(x = 22\)

Answer: 22 students listened to live music only.

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Example 20;

The sets \(A=\{1,3,5,7,9,11\}\), \(B=\{2,3,5,7,11,15\}\) and \(C=\{3,6,9,12,15\}\) are subsets of \(\varepsilon =\{1,2,3,5,6,7,9,11,12,13,15\}\).

a) Draw a Venn Diagram to illustrate the given information.
b) Use your diagram to find:
   i) \(C \cap A'\)
   ii) \(A' \cap (B \cup C)\)

Solution;

(a) Venn Diagram

 

(b) Set Operations:

i) Find \(C \cap A'\)

First, identify \(A'\) (Elements in the universal set \(\varepsilon\) not in A):

\(A' = \{2, 6, 12, 15\}\)

Now, find the intersection with C:

\(C \cap A' = \{6, 12, 15\}\)

ii) Find \(A' \cap (B \cup C)\)

First, find the union \((B \cup C)\):

\((B \cup C) = \{2, 3, 5, 6, 7, 9, 11, 12, 15\}\)

Now, find the intersection of \(A'\) and \((B \cup C)\):

\(A' \cap (B \cup C) = \{2, 6, 12, 15\}\)

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Example 21:

Out of 400 students in the final year in a secondary school, 300 are offering Biology and 190 are offering Chemistry.

i) How many students are offering both Biology and Chemistry if only 70 students are offering neither Biology nor Chemistry?
ii) How many students offer at least one of Biology or Chemistry?

Solution;

 

Let \(n(B)\) = Students offering Biology = 300
Let \(n(C)\) = Students offering Chemistry = 190
Let 70 students offer neither biology nor chemistry.
Universal set \(U = 400\)

i) To find students offering both (x):

The sum of all regions must equal the universal set (\(U = 400\)):

\((300 - x) + x + (190 - x) + 70 = 400\)
\(300 + 190 - x + 70 = 400\)
\(560 - x = 400\)
\(x = 560 - 400\)
\(x = 160\)

ii) To find students offering at least one (Biology or Chemistry):

This is represented by the Union (\(B \cup C\)):

\(n(B \cup C) = U - n(\text{neither})\)
\(n(B \cup C) = 400 - 70\)
\(n(B \cup C) = 330\)

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Example 22;

In a class of 55 students, 21 study Physics, 24 study Geography and 23 study Economics. 6 study both Physics and Geography, 8 study both Geography and Economics, and 5 study both Economics and Physics. If \(x\) study all the 3 subjects and \(2x\) study none of the three subjects, find:
(i) The value of \(x\), (ii) Physics only, (iii) Only two subjects.

Solution;

 

Given Parameters:

• \(n(U) = 55\)
• \(n(P) = 21, n(G) = 24, n(E) = 23\)
• Intersections: \(PG=6, GE=8, EP=5, \text{All}=x, \text{None}=2x\)

Method 1: The Step-by-Step Region Method

This involves defining every single piece of the Venn diagram separately.

1. Isolate "Only" Regions:
Physics Only \(= 21 - (6-x+x+5-x) = \mathbf{10+x}\)
Geography Only \(= 24 - (6-x+x+8-x) = \mathbf{10+x}\)
Economics Only \(= 23 - (8-x+x+5-x) = \mathbf{10+x}\)

2. Sum of all 8 regions:
\((10+x) + (10+x) + (10+x) + (6-x) + (5-x) + (8-x) + x + 2x = 55\)
\(49 + 3x = 55 \implies 3x = 6 \implies \mathbf{x = 2}\)

### OR ###

Method 2: General Inclusion-Exclusion Formula

\(n(U) = [n(P)+n(G)+n(E)] - [\text{Sum of Double Intersections}] + n(\text{All}) + \text{None}\)

\(55 = (21 + 24 + 23) - (6 + 8 + 5) + x + 2x\)
\(55 = 68 - 19 + 3x\)
\(55 = 49 + 3x \implies \mathbf{x = 2}\)

Final Answers:

(i) Value of \(x = \mathbf{2}\)
(ii) Physics only \(= 10 + x = 10 + 2 = \mathbf{12}\)
(iii) Only two subjects \(= (6-x) + (8-x) + (5-x)\)
\(= (6-2) + (8-2) + (5-2) = 4 + 6 + 3 = \mathbf{13}\)

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Example 23;

In a Senior Secondary School, 80 students play Hockey or Football. The number of students that play Football is 5 more than twice the number that play Hockey. If 15 students play both games and every student in the school plays at least one game:

I. Draw a Venn Diagram to illustrate this information.
II. Use the General Set Formula to find the number of students.

Solution;

Given Data:

• Universal Set (\(U\)) / Union (\(F \cup H\)) = 80
• Both games \(n(F \cap H) = 15\)
• Let Hockey \(n(H) = H\)
• Football \(n(F) = 2H + 5\)

Method 1: The Venn Diagram Method

Football only: \(n(F) - 15 = (2H + 5) - 15 = \mathbf{2H - 10}\)
Hockey only: \(\mathbf{H - 1 5}\)
Equation: \((2H - 10) + 15 + (H - 15) = 80\)
\(3H - 10 = 80 \implies 3H = 90 \implies \mathbf{H = 30}\)

### OR ###

Method 2: The General Formula

\(n(F \cup H) = n(F) + n(H) - n(F \cap H)\)

\(80 = (2H + 5) + H - 15\)
\(80 = 3H - 10\)
\(90 = 3H \implies \mathbf{H = 30}\)

Final Results:

a) Total number of students that play Football:
\(n(F) = 2(30) + 5 = \mathbf{65}\)

b) Number of students that play Football but not Hockey:
\(2(30) - 10 = \mathbf{50}\)

c) Number of students that play Hockey but not Football:
\(30 - 15 = \mathbf{15}\)

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Example 23;

In a Senior Secondary School, 80 students play Hockey or Football. The number of students that play Football is 5 more than twice the number that play Hockey. If 15 students play both games and every student in the school plays at least one game:

I. Draw a Venn Diagram to illustrate this information.
II. Use the General Set Formula to find the number of students.

Solution;

Given Data:

• Universal Set (\(U\)) / Union (\(F \cup H\)) = 80
• Both games \(n(F \cap H) = 15\)
• Let Hockey \(n(H) = H\)
• Football \(n(F) = 2H + 5\)

Method 1: The Venn Diagram Method

Football only: \(n(F) - 15 = (2H + 5) - 15 = \mathbf{2H - 10}\)
Hockey only: \(\mathbf{H - 1 5}\)
Equation: \((2H - 10) + 15 + (H - 15) = 80\)
\(3H - 10 = 80 \implies 3H = 90 \implies \mathbf{H = 30}\)

### OR ###

Method 2: The General Formula

\(n(F \cup H) = n(F) + n(H) - n(F \cap H)\)

\(80 = (2H + 5) + H - 15\)
\(80 = 3H - 10\)
\(90 = 3H \implies \mathbf{H = 30}\)

Final Results:

a) Total number of students that play Football:
\(n(F) = 2(30) + 5 = \mathbf{65}\)

b) Number of students that play Football but not Hockey:
\(2(30) - 10 = \mathbf{50}\)

c) Number of students that play Hockey but not Football:
\(30 - 15 = \mathbf{15}\)

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Example 23;

At a Luncheon party for 37 eminent personalities, 25 requested for fried rice, 15 requested for salad while 20 requested for Shredded beef. 7 people requested for fried rice and Salad, 13 people requested for fried rice and Shredded beef and 8 people requested for Salad and Shredded beef. Also, 10 people requested for fried rice only, while 5 people requested for salad only. Using a venn diagram, find; i. The nunmber of people that requested for all the three dishes; ii. The number of people that requested for two different dishes: iii. The number of people that requested for only one dish.

Solution;

Given Data:

• Total \(U = 37\)
• Fried Rice \(n(F) = 25\), Salad \(n(S) = 15\), Shredded Beef \(n(B) = 20\)
• Intersections: \(FS = 7, FB = 13, SB = 8\)
• Fried Rice Only = 10, Salad Only = 5

 

i) Solve for \(x\) (All Three Dishes)

Using the Fried Rice circle (\(F\)) to find the intersection of all three:

\(25 = 10 + (13 - x) + x + (7 - x)\)
\(25 = 30 - x\)
\(x = 30 - 25 \implies \mathbf{x = 5}\)

### OR USE GENERAL FORMULA ###

\(n(F \cup S \cup B) = [n(F)+n(S)+n(B)] - [n(FS)+n(FB)+n(SB)] + n(FSB)\)

\(37 = (25 + 15 + 20) - (7 + 13 + 8) + x\)
\(37 = 60 - 28 + x\)
\(37 = 32 + x \implies \mathbf{x = 5}\)

Final Calculations:

ii) Number of people that requested for TWO different dishes:
Sum of intersections (excluding the center):
\((7 - 5) + (8 - 5) + (13 - 5) = 2 + 3 + 8 = \mathbf{13}\)

iii) Number of people that requested for ONLY ONE dish:
\(n(F) \text{ only} = 10\)
\(n(S) \text{ only} = 5\)
\(n(B) \text{ only} = 20 - [(8-5) + 5 + (13-5)] = 20 - 16 = 4\)
Total: \(10 + 5 + 4 = 19\)

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Example 24;

\(U = \{m, a, r, g, i, n, e, t, c\}\)
\(X = \{a, r, g, e, n, t\}\)
\(V = \{m, a, r, g, i, c\}\)
\(Z = \{e, n, i, g, m, a\}\)

List the members of:

i) \(X \cup V \cup Z\)

ii) \((X \cup V \cup Z)'\)

iii) \(X \cap V \cap Z\)

iv) \(X \cap V' \cap Z\)

v) \(X \cap V \cap Z'\)

vi) \(X' \cap V \cap Z\)

vii) \(X \cap V' \cap Z'\)

viii) \(X' \cap V \cap Z'\)

Solution;

• i) \(X \cup V \cup Z\): \(\{m, a, r, g, i, n, e, t, c\}\) (The entire Universal set)
• ii) \((X \cup V \cup Z)'\): \(\{\}\) or \(\emptyset\) (Empty set)
• iii) \(X \cap V \cap Z\): \(\{a, g\}\)
• iv) \(X \cap V' \cap Z\): \(\{e, n\}\)
• v) \(X \cap V \cap Z'\): \(\{r, t\}\)
• vi) \(X' \cap V \cap Z\): \(\{i\}\)
• vii) \(X \cap V' \cap Z'\): \(\{\}\) (Empty set)
• viii) \(X' \cap V \cap Z'\): \(\{m, c\}\)

A ∪ B

∅ ⊆ S

Set Theory Interactive Lab

Deepen your understanding of the notes above. Ask for proofs, Venn diagrams, or practice problems.

Try asking these based on the notes:

"Show me a 3-set Venn Diagram" "Explain Power Sets with an example" "What is the Cardinality of an empty set?"

AI Specialist: Set Theory & Discrete Math