Mastering Set Theory: From Venn Diagrams to Power Sets & Cardinality

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.

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