MATHEMATICS O LEVEL(FORM TWO) NOTES – SETS

SETS

A set is a group or collection of things such as a herd of cattle, a pile of books, a collection of trees, a swarm of bees, and a flock of sheep.

Description of sets

A set is described or denoted by curly brackets { } and named by capital letters.

Examples

If A is a set of books in the library, then A is written as:

A = {All books in the library}

and read as “A is a set of all books in the library”.

The things or objects in the set are called elements or members of the set.

Example

1. If John is a student of class B, then John is a member of class B and is shortly denoted as ∈ B.
2. If A = {1, 2, 3} then 1 ∈ A, 2 ∈ A, and 3 ∈ A.

The number of elements in a set is denoted by n(A).

Example

1. If A = {a, e, i, o, u} then n(A) = 5.

Describe the following set by:

1. Words
2. Listing
3. Formula

Solution:

1. By words:
   A = {even numbers}
2. By listing:
   A = {2, 4, 6, 8, …}
3. By formula:
   A = {x : x = 2n} where n = {1, 2, 3, …} and is read as “A is a set of all elements x such that x is an even number”.

2. Describe the following sets by listing:

A = {whole numbers between 1 and 8}

Solution:

A = {2, 3, 4, 5, 6, 7}

3. Write the following sets in words:

A = {an integer < 10}

Solution:

A = {integers less than ten} or “A is a set of integers less than ten”.

TYPES OF SETS

Finite set: A set where all elements can be counted exhaustively.

Infinite set: A set whose elements cannot be exhaustively counted.

Example:

B = {2, 4, 6, 8, …}

Empty set: A set with no elements. An empty set is denoted by { } or Ø.

Example:

If A is an empty set, then A = { } or A = Ø.

Exercise

1. List the elements of the named sets:
   • A = {x : x is an odd number < 10} → A = {1, 3, 5, 7, 9}
   • B = {days of the week which begin with letter S} → B = {Saturday, Sunday}
   • C = {prime numbers less than 13} → C = {2, 3, 5, 7, 11}
2. Write the named sets in words:
   • B = {x : x is an odd number < 12} → “B is a set of x such that x is an odd number less than twelve”.
   • E = {x : x is a student in your class} → “E is a set of x such that x is a student in your class”.
3. Write the named sets using the formula method:
   • A = {all men in Tanzania} → A = {x : x is all men in Tanzania}
   • B = {all teachers in your school} → B = {x : x is all teachers in your school}
   • C = {all regional capitals in Tanzania} → C = {x : x is all regional capitals in Tanzania}
   • D = {b, c, d, f, g, …} → D = {x : x is a consonant}

COMPARISON OF SETS

Sets may be equivalent, equal, or one may be a subset of the other.

Equivalent sets are sets whose members match exactly in number.

Example

A = {2, 4, 6, 8} and B = {a, b, c, d}

Then A and B are equivalent sets because the two sets can be matched one-to-one.

Generally, if n(A) = n(B), then A and B are equivalent sets.

Example

If A = {1, 2, 3, 4} and B = {1, 2, 3, 4}, since n(A) = n(B) and the elements are alike, then set A is equal to set B.

Subset

Given two sets A and B, B is said to be a subset of A if all elements of B belong to A.

Example

If A = {a, b, c, d, e} and B = {a, b, c, e}, set B is a subset of A since all elements of B belong to A. But set B has fewer elements than set A. Then set B is a proper subset of set A, and A is a superset of B.

If A = B, then either A is an improper subset of B or B is an improper subset of A.

Symbolically written as A ⊆ B or B ⊆ A.

Note: An empty set is a subset of any set.

The number of subsets in a set is found by the formula 2ⁿ, where n is the number of elements in the set.

Example

1. List all subsets of A = {a, b}.

Solution:

Number of subsets = 2² = 4

The subsets of A are { }, {a}, {b}, {a, b}.

2. How many subsets are there in A = {1, 2, 3, 4}?

Solution:

Number of subsets = 2⁴ = 16

UNIVERSAL SET [U]

A universal set is a single set which contains all elements under consideration. For example, the set of integers contains all elements of sets such as odd numbers, even numbers, counting numbers, and whole numbers. In this case, the set of integers is the universal set.

Exercise

1. Which of the following sets are:
   • a) Finite set
   • b) Infinite set
   • c) Empty set

• A = {Nairobi, Dar es Salaam}
• B = {2, 4, 6, …, 36}
• E = {All mango trees in the world}
• F = {x : x is all students aged 100 years in your school}
• H = {1, 3, 5, 7}
• D = {all lions in your school}
• I = Ø

Solution:

• Finite sets: A, B, H
• Infinite sets: E, F
• Empty sets: D, I

2. In each of the following pairs of sets, show by matching whether the pairs are equivalent or not.

A = {a, b, c, d} and B = {b, c, d, e}

Not equivalent:

B = {Rufiji, Ruaha, Malagarasi} and C = {lion, leopard}

B and C are not equivalent.

3. Which of the following sets are equal?

A = {a, b, c, d}, B = {d, a, b, c}, C = {a, e, i, o, u}, D = {a, b, c, d}, E = {d, c, b, a}, F = {a, e, b, c, d}

Solution:

A, B, D, and E are equal.

4. List all subsets of each of the following sets:

1. A = {1}

Number of subsets = 2¹ = 2

Subsets: { }, {1}

1. B = Ø

Number of subsets = { }

1. C = {Tito, Juma}

Number of subsets = 2² = 4

Subsets: { }, {Tito}, {Juma}, {Tito, Juma}

5. Name the subsets of each pair using the symbol ⊂

1. A = {a, b, c, d, e, f, g, h} and B = {d, e, f}
   Therefore, B ⊂ A
2. A = {2, 4} and D = {2, 4, 5}
   Therefore, A ⊂ D
3. A = {1, 2, 3, 4, …} and B = {2, 4, 6, 8, …}
   Therefore, A ⊆ B

6. Given G = {cities, towns, and regions of Tanzania}, which of the following sets are subsets of G?

• A = {Nairobi, Dar es Salaam}
• B = {Dodoma, Mombasa, Mwanza}
• C = { }
• D = {Arusha, Iringa, Bagamoyo}
• E = {Mbeya, Tunduru, Ruvuma}

Therefore, the subsets of G are C, D, and E.

7. Which of the following sets are subsets of K given that K = {p, q, r, s, t, u, v, w}?

• A = {p, s, t, x}
• B = {q, r, d, t}
• C = { }
• D = {p, q, r, s, t, u, v, w}
• E = {a, b, c, d}
• F = {s, v, q}

Therefore, the subsets of K are D, C, and F.

8. What is n(A) if A = { }?

n(A) = 0

9. Write in words the universal set of the following sets:

1. A = {a, b, c, d}
   The universal set of A is the set of alphabets.
2. B = {1, 2, 3, 4}
   The universal set of B is the set of natural numbers.

OPERATIONS WITH SETS

UNION

The union of two sets A and B is formed when the members of both sets are combined without repetition. The union is denoted by A ∪ B and is defined as:

A ∪ B = {x : x ∈ A or x ∈ B}

Example

1. If A = {2, 4, 6} and B = {2, 3, 5} then A ∪ B = {2, 3, 4, 5, 6}

INTERSECTION

The intersection of two sets A and B is a new set formed by taking common elements. The symbol for intersection is “∩”.

A ∩ B = {x : x ∈ A and x ∈ B}

Example

1. If A = {1, 2, 3, 4, 5} and B = {1, 3, 5} then A ∩ B = {1, 3, 5}
2. Find A ∩ B when A = {a, b, c, d, e, f} and B = {a, e, i, o, u}.
   Solution: A ∩ B = {a, e}

COMPLEMENT OF A SET

If A is a subset of a universal set U, then the members of U which are not in A form the complement of A, denoted by A′ or A^c.

Example

If U = {a, b, c, …, z} and A = {a, b}, then A′ = {c, d, e, …, z}.

Given U = {15, 45, 135, 275} and A = {15}, find A′.
Solution: A′ = {45, 135, 275}

JOINT AND DISJOINT SETS

Joint sets are sets with common elements.

Example: A = {1, 2, 3, 5}, D = {1, 2} then A and D are joint sets since {1, 2} are common elements.

Disjoint sets are sets with no elements in common.

Example: A = {a, b, c} and B = {1, 2, 3, 4} then A and B are disjoint sets since they do not have any common element.

EXERCISE

1. Find:
   1. Union
   2. Intersection

1. A = {5, 10, 15}, B = {15, 20}
   1. A ∪ B = {5, 10, 15, 20}
   2. A ∩ B = {15}
2. A = { }, B = {14, 16}
   1. A ∪ B = {14, 16}
   2. A ∩ B = { }
3. A = {First five letters of the English alphabet}, B = {a, b, c, d, e}
   1. A ∪ B = {a, b, c, d, e}
   2. A ∩ B = {a, b, c, d, e}
4. A = {counting numbers}, B = {prime numbers}
   1. A ∪ B = {counting numbers}
   2. A ∩ B = {prime numbers}

VENN DIAGRAM

Venn diagrams are diagrams (ovals) devised by John Venn for representation of sets.

Example:

If A = {a, b, c} it can be represented as:

µ is the universal set, in this case the set of all English alphabets. If the sets have any elements in common, the ovals overlap. For example, if A = {a, b, c} and B = {a, b, c, d} then it can be represented as:

Disjoint sets can also be represented on a Venn diagram.

Example: If A = {a, b} and B = {1, 2}, the relation between A and B is as follows:

Examples

If A is a subset of B, represent the two sets on a Venn diagram.

Represent A = {2, 3, 5}, B = {2, 5, 7}, C = {2, 3, 7} in a Venn diagram.

Represent A ∪ B in a Venn diagram given that A = {1, 2}, B = {1, 3, 5}.

If sets A and B have the same elements in common, represent the following in a Venn diagram:

1. A and B
2. A and B

Solution:

See diagrams above.

Word Problems

1. In a certain school of 120 students, 40 learn English, 60 learn Kiswahili, and 30 learn both Kiswahili and English. How many students learn:
   1. English only
   2. Neither English nor Kiswahili

Solution:

Let µ = {students in the school}

A = {students learning English}

B = {students learning Kiswahili}

a) Number of students learning English only = n(A) – n(A ∩ B) = 40 – 30 = 10

b) Number of students learning neither English nor Kiswahili = n(µ) – [n(A) + n(B) – n(A ∩ B)] = 120 – [40 + 60 – 30] = 120 – 70 = 50

Alternatively, by Venn diagram, 50 students learn neither English nor Kiswahili.

2. In a certain school, 50 students eat meat, 60 eat fish, and 25 eat both meat and fish. Assuming every student eats meat or fish, find the total number of students in the school.

Solution:

Let µ = {total number of students}

A = {students eating fish}

B = {students eating meat}

n(A ∪ B) = n(A) + n(B) – n(A ∩ B) = 50 + 60 – 25 = 85 students.

Revision Exercise

1. How many subsets are there in A = {a, b, c, d, e, f, g}?

Solution:

Since n(A) = 7, number of subsets = 2⁷ = 128.

Set A has 128 subsets.

2. List all the subsets of A = {2, 4, 6}.

Solution:

n(A) = 3, number of subsets = 2³ = 8.

The subsets are { }, {2}, {4}, {6}, {2, 4}, {2, 6}, {4, 6}, {2, 4, 6}.

3. If µ = {a, b, c, d, e}, B = {e, d}, A = {a, b, c}, list the elements of B′.

a) B′ = {a, b, c}

b) Find A′ and B′.

Solution:

A′ = {d, e}

B′ = {a, b, c}

A′ ∩ B′ = { }

4. Draw a Venn diagram and shade the required region of the following:

1. A′ ∩ B
2. B′ ∩ A
3. A ∩ B

5.

In a group of 29 tourists from different countries, 17 went to Manyara National Park, 13 to Mikumi National Park, and 8 went to neither. How many tourists went to both places?

To find x:

17 – x + x + 13 – x + 8 = 29

38 – x = 29

x = 9

Therefore, 9 tourists went to both places.

6.

From the figure:

1. List the members of set A: A = {1, 2, 5, b}
2. List the members of set C: C = {1, 4, a, b}