Unit 1-Set Theory
Introduction to Sets
Defining a Set
A set is a collection of objects that are clearly identified.
The objects in the set are called the members or elements of the set.
Members belong to the set.
Denoted by capital letters: A, B, C, ..., Z.
Elements are represented by lower case letters: a, b, c, ..., z.
Set Membership Notation
Belonging to a Set
The symbol 'E' denotes belongs to or is an element of a set.
If a ∈ A, it means "a is an element of set A".
If a ∉ A, it means "a is not an element of set A".
The order of elements is unimportant.
Common Properties of Sets
Specifying Common Properties
A common property among items allows the grouping of "things" into a set.
Examples include:
Clothing items: shoes, socks, hat, shirt, pants.
Types of fingers: index, middle, ring, pinky.
Notation of Sets
Curly Bracket Notation
Sets have a simple notation by listing each element separated by commas within curly brackets:
Example: {3, 6, 9, 12}.
Curly brackets are also referred to as "set brackets" or "braces".
Example notation for clothing: {socks, shoes, watches, shirts}.
Examples of Sets
Specific Set Examples
Example 1: The set V of vowels in the English alphabet:
V = {a, e, i, o, u}.
Example 2: The set O of odd positive integers less than 10:
O = {1, 3, 5, 7, 9}.
Numerical Sets (Well-Defined)
Types of Numerical Sets
Set of even numbers: {..., -4, -2, 0, 2, 4, ...}.
Set of odd numbers: {..., -3, -1, 1, 3, ...}.
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}.
Positive multiples of 3 less than 10: {3, 6, 9}.
Well-Defined Set Examples
Specific Example Sets
A = {2, 4, 6, 8, 10, 12, 14} (even numbers between 1 and 15).
B = {10, 15, 20, 25} (multiples of 5 between 8 and 28).
Sets are often represented by capital letters as shown.
Identifying Well-Defined Sets
Your Task on Well-Defined Sets
Determine which of the following are well-defined sets:
All the colors in the rainbow.
All the points on a straight line.
All honest members in the family.
All consonants in the English alphabet.
All tall boys in the school.
All hardworking teachers in a school.
All prime numbers less than 100.
All letters in the word GEOMETRY.
Answers: 1, 2, 4, 7, 8 are well-defined.
Membership of Sets
Examples
Is a member of:
3 ∈ {1, 2, 3}.
-1 ∈ {-5, -1, -7}.
Is not a member of:
3 ∉ {4, 5, 6}.
a ∉ {b, c, d}.
Elements of a Set
Example of Elements
Example Set A of odd numbers between 1 and 10: A = {3, 5, 7, 9}.
Elements: 3, 5, 7, 9 belong to set A.
The number of elements in set A is 4: n(A) = 4.
Representation of a Set
Types of Set Representation
Sets can be represented in two ways:
Roster or Tabular Form.
Set Builder Notation.
Tabular Form
Understanding Tabular Form
Listing all elements separated by commas and enclosed in braces {}.
Example: A = {1, 2, 3, 4, 5} (first five natural numbers).
Example: B = {2, 4, 6, 8, ...} (even numbers up to 50).
Set Builder Form
Understanding Set Builder Form
Writing characteristic properties of elements symbolically.
Example: A = {x | x ≤ 5}.
Example: B = {x | 0 < x < 50}.
Written Tasks
Set Builder Form Tasks
Write sets in set builder form:
(a) A = {2, 4, 6, 8}.
(b) B = {3, 9, 27, 81}.
(c) C = {1, 4, 9, 16, 25}.
(d) D = {1, 3, 5, ...}.
(e) E = {a, e, i, o, u}.
Answers for Set Builder Form
Written Tasks on Set Builder Form
(a) {x | x is even and x < 8}.
(b) {x | x = 3^n, n ∈ N, n < 4}.
(c) {x | x = n², n ≤ 5, n ∈ N}.
(d) {x | x is odd}.
(e) {x | x = vowels in the English alphabet}.
Roster Form Tasks
Written Tasks on Roster Form
Write sets in roster form:
(a) {x | x ∈ W, x < 5}.
(b) {all even numbers less than 12}.
(c) {x | x is divisible by 12}.
(d) {first seven natural numbers}.
(e) {whole numbers less than 5}.
Answers for Roster Form Tasks
Written Tasks on Roster Form
(a) {0, 1, 2, 3, 4, 5}.
(b) {-2, -4, -6, -8, -10, 2, 4, 6, 8, 10}.
(c) {12, 24, 36, ...}.
(d) {1, 2, 3, 4, 5, 6, 7}.
(e) {0, 1, 2, 3, 4}.
Finite and Infinite Sets
Definitions
Finite Set:
Possible to list and count all members.
Examples:
D = {days of the week} → n(D) = 7.
A = {1, 2, 3, 4, 5} → n(A) = 5.
F = {-3, -2, -1, 0, 1, 2, 3} → countable elements.
Infinite Set:
Not possible to list or count all members.
Examples:
E = {even numbers > 9}. n(E) = infinite.
G = {whole numbers > 2000}. n(G) = infinite.
Classification of Finite and Infinite Sets
Your Task
Classify the following:
a. A = { x | x ∈ N and x is even}
b. B = { x | x ∈ N and x is composite}
c. C = { x | x ∈ N and 3x - 2 = 0}
d. D = { x | x ∈ N and x² = 9}
e. E = { multiples of 3}
f. F = { letters in the English alphabet}
g. G = { number of persons in a house}
h. H = {x | x ∈ P, P is a number}
i. I = { fractions with numerator 3}.
Answers: a. Infinite, b. Infinite, c. Finite, d. Finite, e. Infinite, f. Finite, g. Finite, h. Infinite, i. Infinite.
Empty Set / Null Set
Definitions
An empty set has no members, denoted by {} or ∅.
Example:
H = {number of dinosaurs on Earth} → empty set.
Note: 0 ≠ {0}; the left side is empty, the right is a singleton set.
Singleton Set or Unit Set
Definition
A singleton is a set that contains exactly one element.
Denoted as {a}.
Example: S = {x | x ∈ N, 7 < x < 9}.
Identifying Null and Singleton Sets
Your Task
Identify sets as null or singleton:
a. A = {x | x ∈ N, 1 < x < 2}
b. P = {intersection of two lines}
c. C = { x | x is an even prime number > 2}
d. {x | x is an even prime number}
e. E = {x | x² = 9, x is even}
f. B = {0}
g. D = {largest 1-digit number}
h. F = {triangles with 4 sides}
i. H = {even numbers not divisible by 2}.
Answers: a. Null, b. Singleton, c. Null, d. Singleton, e. Null, f. Singleton, g. Singleton, h. Null, i. Null.
Equal and Equivalent Sets
Definitions
Equal Sets: Two sets are equal if they have the same members.
Examples:
A = {1, 2, 3} and B = {1, 2, 3}. A = B.
C = {1, 2, 5} and D = {5, 1, 2}. C = D.
Equivalent Sets: Two sets are equivalent if they have the same number of elements.
Examples:
F = {2, 4, 6, 8, 10} and G = {10, 20, 30, 40, 50}. n(F) = n(G).
A = {1, 2, 3} and B = {a, b, c}. n(A) = n(B).
Disjoint Sets
Definitions
Disjoint Sets: Two sets are disjoint if they have no elements in common.
Example: S = {2, 4, 6, 8} and T = {1, 3, 5, 7}.
Their intersection is the empty set: S ∩ T = ∅.
Union of Sets
Definitions
Union: Combining all elements of two sets.
Represented as A ∪ B.
Example: If A = {1, 2, 3, 4, 5} and B = {2, 4, 6}, then A ∪ B = {1, 2, 3, 4, 5, 6}.
Intersection of Sets
Definitions
Intersection: The set of common elements of two or more sets.
Denoted by the symbol ∩.
Example: If A = {1, 2, 3, 7, 11} and B = {1, 4, 7, 10}, then A ∩ B = {1, 7}.
Difference of Sets
Definitions
Difference Set: Members of set A that are not in set B.
Example: A = {0, 1, 2, 3}, B = {2, 3}. A - B = {0, 1}.
Universal Set
Definition
The universal set includes all elements considered within a specific context.
Denoted by U.
Example: U = {6, 7, 8, 9, 15, 16, 17, 18, 20, 21}.
Complement of a Set
Definition
If A is a subset of universal set U, then U - A or U \ A represents the complement of A.
Example: If U = {3, 5, 7, 9, 11, 13, 15, 17, 19} and A = {5, 11, 17, 19}, then U - A = {3, 7, 9, 13, 15}.
Symmetric Difference
Definition
The symmetric difference of A and B, denoted by A ⊕ B, is the set of (A - B) ∪ (B - A).
Example: U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}. A ⊕ B = {1, 2, 3, 6, 7, 8}.
Cartesian Product
Example of Cartesian Products
Given sets A = {0, 1} and B = {a, b, c}:
A × B = {(0, a), (0, b), (0, c), (1, a), (1, b), (1, c)}.
Set Properties
Commutative Laws
A ∩ B = B ∩ A
A ∪ B = B ∪ A
Associative Laws
(A ⊕ B) ⊕ C = A ⊕ (B ⊕ C)
(A ∪ B) ∪ C = A ∪ (B ∪ C)
Distributive Laws
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
De Morgan's Laws
(A ∩ B)' = A' ∪ B'
(A ∪ B)' = A' ∩ B'
Subsets
Definition
A set A is a subset of set B if all elements of A are also contained in B.
Denoted as A ⊆ B.
The empty set is a subset of every set.
Examples of Subsets
Specific Examples
If B = {3, 5, 6, 8, 9, 10, 11, 13} and A = {5, 11, 13}, then A ⊆ B.
Subsets include: {5, 11, 13}, {3, 5}, and the empty set {}. Not a subset: {1, 6} (contains 1 which is not in B).
Proper and Improper Subsets
Definitions
An improper subset includes the set itself.
Example: For A = {1, 2, 3}, then {1, 2, 3} is an improper subset of A.
A proper subset does not include the entire set.
Cardinality of Sets
Definition
Cardinality of a set S, denoted |S|, is the number of elements in S.
Case Examples:
|A| = 0 for an empty set.
|A| = n if A has n elements.
|A| = ∞ if A is infinite.
Examples of Cardinality
Specific Cases
Let A = {odd positive integers < 10}, then |A| = 5.
Let S = {letters in English alphabet}, then |S| = 26.
Let P = {infinite numbers}, then |P| = ∞.