DEMO

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.

Examples:

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…” xAx is in the set A
“…is not an element of…” xAx 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
AB Subset A is a subset of B
AB Proper subset A is a proper subset of B
AB 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.

Example:

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 ϕ .  
  • If A is a null set then n(A) = 0.
Examples:

1.

If A is a set of multiples of 10 between 0 and 9.
A = { } /ϕ

n(A) = 0

2.

B is a set of chicken with 4 legs.

B = { } /

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.

If D = {2; 4; 6; 8} and E = {4;8;5;2}, then D  E.

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 (υ)

  • The union of two sets is the set which contains all the elements of the other sets.
Example:
If P = {r; e; s; t} and Q = {p; l; a; y}
Then P  Q = {r; e; s; t; p; l; a; y}
n(P  Q) = 8


Diagrams:

Intersection (∩)

  • The intersection of two sets is a set of elements that are common to both sets.
Examples:
Given that X = {3; 5; 7; 9; 11} and Y = {5; 7; 11; 17; 19}
Then X  Y = {5; 7; 11}
n(X  Y) = 3

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:

  • F G
  • F G
  • n(F)
  • n(G)
  • Draw a Venn diagram showing sets F and G.

Solution
  1. F = {1; 2; 3; 4; 5} and G = {1; 3; 5; 7; 8; 9}
  2. FG = {1; 2; 3; 4; 5; 7; 8; 9}
  3. F  G = {1; 3; 5}
  4. n(F) = 5
  5. n(G) = 6
  6. Venn diagram

2.

ξ= {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:
    • n(A B) =
    • n(B A) =
    • n( ξ ) =
  • Show ξ , A and B on a Venn diagram.

Solution

a. Using set symbols

  • A ξ
  • h A
  • A B
  • ξ B

b.

  • n A B = 7
    • A B = { a ; b ; c ; d ; f ; h ; i }
  • n B A = 2
    • B A = { b ; c }
  • n ξ = 10

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

      PL=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