Sets – SS1 Further Mathematics Lesson Note (First Term)

A set is a well-defined collection of objects or numbers, where it is always clear whether an item belongs to the collection or not.

• Examples: a collection of books, a set of cooking utensils, the days of the week.
• Naming sets: use capital letters, such as P, Q and R.
• Naming elements: use small letters, such as a, b and c.
• Writing a set: put the elements inside curly brackets and separate them with commas.

Behavioural objectives

By the end of the lesson, students should be able to:

1. define a set and list its elements, including sets written in set notation;
2. state the cardinality of a set;
3. identify finite, infinite, empty, disjoint, universal and complementary sets, and subsets and supersets;
4. find the union, intersection and complement of sets;
5. use n(A ∪ B) = n(A) + n(B) – n(A ∩ B) to solve problems;
6. use Venn diagrams to solve problems involving two sets.

What are the common terms used in sets?

The common terms in sets are element, cardinality and set notation.

What are the elements of a set?

The elements of a set are its members, written inside curly brackets { } and separated by commas.

• Example: in M = {1, 3, 5, 7, 11}, 1 is an element of M.
• Membership symbol: ∈ means “is an element of”, so 3 ∈ M.
• Non-membership symbol: ∉ means “is not an element of”, so 4 ∉ M.

Example: Write down the elements in each of the following sets.

1. A = {odd numbers from 1 to 21}
2. F = {factors of 30}
3. M = {multiples of 4 up to 40}

Solution:

1. A = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21}
2. F = {1, 2, 3, 5, 6, 10, 15, 30}
3. M = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40}

What is the cardinality of a set?

The cardinality of a set is the number of elements in the set, and the cardinality of A is written n(A).

• Example: if A = {2, 4, 6}, then n(A) = 3.

Example: Given that µ = {all the days of the week} and B = {all days of the week whose names begin with S}:

1. List all the elements of µ.
2. List the members of B.
3. What is n(µ)?
4. What is n(µ) + n(B)?

Solution:

1. µ = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
2. B = {Sunday, Saturday}
3. n(µ) = 7
4. n(µ) + n(B) = 7 + 2 = 9

What is set notation?

Set notation describes a set by a rule, using inequality and other symbols, instead of listing every element.

• Example: B = {x: -10 ≤ x ≤ 3, x is an integer}.
• How to read it: “B is the set of all x such that x is an integer from -10 to 3.”
• Colon: the symbol : means “such that”.

Example: List the members of the following sets, where x is an integer.

1. A = {x: 5 < x < 8}
2. B = {x: 0 ≤ x ≤ 5}

Solution:

1. A = {6, 7}, because 5 and 8 are not included.
2. B = {0, 1, 2, 3, 4, 5}, because 0 and 5 are included.

What are the common set symbols?

What are the types of sets?

There are seven types of sets in this topic: finite, infinite, empty, subset and superset, disjoint, universal and complementary sets.

1. Finite set: a set in which all the members can be listed, such as {1, 2, 3, 4}.
2. Infinite set: a set in which all the members cannot be listed, such as the set of counting numbers.
3. Empty (null) set: a set without any member, written { } or Ø.
4. Subset: A is a subset of B if all the elements of A are contained in B. It is written A ⊂ B.
5. Superset: if A ⊂ B and B has at least one element not in A, then B is a superset of A.
6. Disjoint sets: two sets with no common element, that is, no intersection. We write A ∩ B = Ø.
7. Universal set: the set that contains all the members under consideration in a given problem. It is written µ or ξ.
8. Complementary set: the set of members of the universal set that are not in A. It is written A′ or A^c.

• Example of subsets: the subsets of {a, b} are { }, {a}, {b} and {a, b}.
• Number of subsets: a set with n elements has 2ⁿ subsets, so {a, b} has 2² = 4.
• Example of disjoint sets: {2, 4, 6} and {1, 3, 5} have no common element.

What are union and intersection of sets?

Union joins all the members of two sets, while intersection picks only the members they share.

What is the intersection of sets?

The intersection of sets is the set of all the common elements in two or more given sets, written ∩.

• Example: if A = {1, 2, 3} and B = {2, 3, 4}, then A ∩ B = {2, 3}.

What is the union of sets?

The union of sets is the set of all members that belong to A or B or both, written ∪.

• Example: if A = {1, 2, 3} and B = {2, 3, 4}, then A ∪ B = {1, 2, 3, 4}.
• Note: a common element is listed once only.

Example: The universal set is µ = {x: 1 ≤ x ≤ 12, x is an integer}. Its subsets D, F and G are D = {x: 2 < x < 8}, F = {x: 4 ≤ x ≤ 10} and G = {x: 1 < x ≤ 4}. Find:

1. D ∪ F
2. D ∩ F
3. G′
4. (D ∩ G)′

Solution:

µ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
D = {3, 4, 5, 6, 7}
F = {4, 5, 6, 7, 8, 9, 10}
G = {2, 3, 4}

1. D ∪ F = {3, 4, 5, 6, 7, 8, 9, 10}
2. D ∩ F = {4, 5, 6, 7}
3. G′ = {1, 5, 6, 7, 8, 9, 10, 11, 12}
4. First find D ∩ G = {3, 4}. Then (D ∩ G)′ = {1, 2, 5, 6, 7, 8, 9, 10, 11, 12}

How are union and intersection of sets related?

The number of elements in a union is given by n(A ∪ B) = n(A) + n(B) – n(A ∩ B).

• In words: the number in A or B equals the number in A plus the number in B, minus the number in both.
• Why subtract: the elements in A ∩ B are counted twice, so one count is removed.

Example: If n(A) = 23, n(B) = 12 and n(A ∪ B) = 35, find n(A ∩ B) and comment on sets A and B.

Solution:

n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
35 = 23 + 12 – n(A ∩ B)
35 = 35 – n(A ∩ B)
n(A ∩ B) = 35 – 35 = 0

Sets A and B are disjoint, because they have no common element.

What is a Venn diagram?

A Venn diagram is a geometric representation of sets that shows the relationships between two or more sets.

• Rectangle: represents the universal set µ.
• Circles: each circle represents one set inside µ.
• Overlap: the shared part of two circles represents the intersection.
• Uses: intersection, union and complement can all be shown on Venn diagrams.

How do you use Venn diagrams to solve problems involving two sets?

Draw two overlapping circles in a rectangle, let the overlap be x, and add all the regions to equal n(µ).

1. Let x stand for the number in both sets.
2. Write “first set – x” and “second set – x” for the parts outside the overlap.
3. Write the number in neither set outside the circles.
4. Add all the regions and set the sum equal to n(µ).
5. Solve for x, then answer the question asked.

Example 1: Biology and Chemistry

Out of the 400 final-year students in a secondary school, 300 are offering Biology and 190 are offering Chemistry. Only 70 students are offering neither Biology nor Chemistry. How many students are offering:

1. both Biology and Chemistry?
2. at least one of Biology or Chemistry?

Solution:

(i) Let the number of students who offer both Biology and Chemistry be x, that is, n(B ∩ C) = x. From the information given:

n(µ) = 400
n(B) = 300
n(C) = 190
n(B ∪ C)′ = 70

The sum of the elements in all the regions equals the total number of elements in the universal set. So:

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

The number of students who offer both Biology and Chemistry is 160.

(ii) The students who offer at least one subject are those who offer Biology only, Chemistry only and both. From the Venn diagram:

(300 – x) + x + (190 – x) = 490 – x
= 490 – 160 (from (i) above)
= 330

So 330 students offer at least one of Biology or Chemistry.

Example 2: Modern and traditional music

In a youth club with 94 members, 60 like modern music and 50 like traditional music. The number of members who like both traditional and modern music is three times the number who do not like any type of music. How many members like only one type of music?

Solution:

Let the number of members who do not like any type of music be X. Then:

n(T ∩ M) = 3X
n(µ) = 94
n(M) = 60
n(T) = 50
n(M ∪ T)′ = X

The sum of the elements in all the regions equals the total number of elements in the universal set. So:

(60 – 3X) + 3X + (50 – 3X) + X = 94
110 – 2X = 94
16 = 2X
X = 8

The members who like only one type of music are those who like modern music only plus those who like traditional music only:

(60 – 3X) + (50 – 3X) = 110 – 6X
= 110 – 6 × 8 (from above)
= 110 – 48
= 62

So 62 members like only one type of music.

How do you use the four regions of a two-set Venn diagram?

A two-set Venn diagram has four regions, and their numbers add up to n(µ).

• Region 1: A ∩ B′, the elements in A only.
• Region 2: A ∩ B, the elements in both A and B.
• Region 3: A′ ∩ B, the elements in B only.
• Region 4: (A ∪ B)′, the elements in neither A nor B.

Therefore, µ = 1 + 2 + 3 + 4, that is:

n(µ) = n(A ∩ B′) + n(A ∩ B) + n(A′ ∩ B) + n(A ∪ B)′

Example: In a class of 40 students, every student had to study French, Russian or both subjects. 25 students studied French and 20 studied Russian. Find the number of students who studied both languages.

Solution:

Let µ = {all the students}, F = {French students} and R = {students studying Russian}.

n(µ) = 40, n(F) = 25, n(R) = 20
n(F ∩ R) = x
n(F ∩ R′) = 25 – x
n(R ∩ F′) = 20 – x

Every student studied at least one language, so no student is outside both circles.

n(µ) = (25 – x) + x + (20 – x)
40 = 45 – x
x = 45 – 40
x = 5

So n(F ∩ R) = 5. Five students studied both languages.

Evaluation

1. List the elements of P = {x: 2 < x < 9, x is an integer}.
2. Define a subset and a disjoint set. Give one example of each.
3. If n(A) = 20, n(B) = 15 and n(A ∩ B) = 6, find n(A ∪ B).
4. In a class of 30 students, every student offers Yoruba, Igbo or both. 18 students offer Yoruba and 20 offer Igbo. How many students offer both languages?
5. Given µ = {0, 1, 2, 3, …, 12}, B = {4, 6, 8, 10, 12} and C = {y: 1 < y < 8, y is a prime number}, list C and find B ∪ C and B ∩ C.

Assignment

In a senior secondary school, 90 students play hockey or football. The number that play football is 5 more than twice the number that play hockey. If 5 students play both games and every student in the school plays at least one of the games, find:

1. the number of students that play football;
2. the number of students that play football but not hockey;
3. the number of students that play hockey but not football.