 # SETS

By the end of the subtopic learners should be able to:
• define sets by listing element or describing the set.
• correctly use set symbols.
• use Venn diagrams to display sets and to solve problems involving no more than two sets.

### Sets

• A set is a collection of objects.
• A set contains members which maybe common in characteristics or any things defined to be members of the set.
• Examples can be a set of types of birds, a set of cooking utensils, a set of fruits and a set of vegetables.

#### 1.

##### If M is a set of all odd numbers between 0 and 12.

M = {odd numbers between 0 and 12}

M = {1;3;5;7; 9;11}

#### 2.

##### If X is a set of the colors of the rainbow.

X = {colors of the rainbow}

X = {red; orange; yellow; green; blue; indigo; violet}

#### 3.

##### V is a set of all vowels in the alphabetical order.

V = {vowels}

V = {a; e; i; o; u}

N.B. Members of the set are contained is special brackets called set braces as shown in the given examples.

### Symbols or notations used in sets

Notation whAT IT REPRESENTS WHat it means
∅ or {} Null set or empty set A set with no elements
ε Universal set A set which consists of all elements
A∪B Union of two sets All elements in A and B collectively
A∩B Intersection of two sets Common elements in both A and B
n(A) Number of elements in a set Number of elements in set A
“…is an element of…”
“…is not an element of…” $\mathrm{x}\notin \mathrm{A}\to$x is not in the set A
A’ Complement of set A All the other elements in the universal set that are not in the set A
$\mathrm{A}\subseteq \mathrm{B}$ Subset A is a subset of B
$\mathrm{A}\subset \mathrm{B}$ Proper subset A is a proper subset of B
$\mathrm{A}\overline{)\subset }\mathrm{B}$ Not a proper subset A is not a proper subset of B
A=B Equal sets A and B have exactly the same elements

### Number of elements in a set

• n(A) is the notation for the number of elements in set A.

#### 1.

##### If S is a set of multiples of 5 from 0 to 35.

S = {5; 10; 15; 20; 25; 30; 35}

n(S) = 7

#### 2.

##### If P is a set of days in a week.

P= {Monday; Tuesday; Wednesday; Thursday; Friday; Saturday; Sunday}

n(P) = 7

#### 3.

##### If T is a set of all months in a year.

T = {January; February; March; April; May; June; July; August; September; October; November; December}

n(T) = 12

### Types of sets

#### Empty sets

• An empty set is also known as a null set.
• Null set can be written as { } or $\varphi .$
• If A is a null set then n(A) = 0.

#### 1.

##### If A is a set of multiples of 10 between 0 and 9.
A = { } /$\varphi$

n(A) = 0

#### 2.

##### B is a set of chicken with 4 legs.

$\mathrm{B}=\left\{\right\}/\varnothing$

n(B) = 0

#### Infinite sets

• An infinite set is a set with an unending list of members.
• If X is a set of naturals then X = {1; 2; 3; 4; 5; 6; …}
• Y is a set of multiples of 2 then Y = {2; 4; 6; 8; 10; …}

#### Subsets

• A subset is a set that is completely contained in another set. ##### Example
1. If X is a set of natural numbers from 10 to 20. Y is a set of odd numbers between 10 and 20.

X = {10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20}
Y = {11; 13; 15; 17; 19}

All the elements of Y are contained in X therefore Y is a subset of X. #### Equal sets

Two sets are equal when they contain exactly the same members or elements, even if they may not be in the same order.

##### Examples
A = {s;e;t;s}
B = {t;e;s;s}

Set A is equal to set B. This is represented as: A = B.

#### Equivalent sets

Two sets are equivalent if they contain the same number of elements.

##### Examples

P = {10;20;30;40;50;60}
S = {11;22;33;44;55;66}
Set P is equivalent to set S.

#### Union and Intersection

Union ($\mathrm{\upsilon }$)

• The union of two sets is the set which contains all the elements of the other sets.

##### Diagrams:   #### Intersection (∩)

• The intersection of two sets is a set of elements that are common to both sets.

##### Diagrams:   In the diagram the region painted red represents the intersection of the two sets whilst the regions painted red, blue and purple put together represent the union of the two sets.

### Venn diagrams

• The diagrammatical representation of sets is what is known as the Venn diagrams.

#### a. #### b. #### c. #### d. #### e. ### Examples and solutions

#### 1.

Given that F = {1; 2; 3; 4; 5} and G = {1; 3; 5; 7; 8; 9}, write down:

6. ##### Venn diagram #### 2.

##### $\mathrm{\xi }$= {a; b; c; d; e; f; g; h; i; j}, A = {a; b; c; d} and B = {b; c; f; h; i}.
• Use set symbols to represent the following statements:
• Set A is a subset of the universal set.
• h is an element of set A.
• Set A is not equal to set B.
• The universal set contains set B.
• Write down the following:

#### C. ### Word problems involving Venn diagrams

1. In a school meeting 10 people have phones, 16 have laptops and 8 have neither of the two. If the total number of people in the meeting is 30, calculate;
• The value of x, the number of people who have both the phone and the laptop.
• The number of people with laptops only.
2. Class of 40 pupils had two tests. 36 of them passed Mathematics and 24 passed English. Calculate;
• The value of x, the number of pupils who passed both subjects.
• Hence draw a Venn diagram showing this information.

##### Solutions
1. If x is the number of people with both the phone and laptop then

$\mathrm{P}\cap \mathrm{L}=\mathrm{X}$

• To get the number of people who have phones or laptops only, we subtract x from 10 and 16 respectively (see diagram below).
• 8 is the number of people who have neither a phone nor laptop and therefore is outside of sets P and L but inside the .
• To get the value of x we add the expressions in the Venn diagram and the total should give us 30 (the number of people in the meeting).
• a. 10 – x + x + 16 – x + 8 = 30
34 - 30 = x
x = 4 Now that we have the value of x we are now able to fill in the actual numbers of elements in the regions of the Venn diagram.(Note that, 8 + 6 + 4 + 12 = 30, the total number of people in the meeting)

2. From the Venn diagram the number of people with laptops only are 12.
1. 36 – x + x + 24 – x = 40

60 – x = 40
x = 20

1. The value of x = 20.
2. The Venn diagram   