Topic 10: Sets – Mathematics Notes Form Two

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WHAT IS SET?

Forget everything you know about number and forget that you even know what a number is. This is where mathematics starts. Instead of mathematics with numbers we will think about mathematics with things.

The word set means collection of related things or objects. Or, things grouped together with a certain property in common. For example, the items you wear: shoes, socks, hat, shirt, pants and so on.

This is called a set. A set notation is simple, we just list each element or member (element and member are the same thing), separated by comma, and then put some curly brackets around the whole thing. See an example below:

The Curly brackets are sometimes called “set brackets” or “Braces”.

Sets are named by capital letters. For example; A = {1, 2, 3, 4 …} and not a = {1, 2, 3, 4, …}.
To show that a certain item belongs to a certain set we use the symbol ∈.
For example if set A = {1, 2, 3, 4} and we want to show that 1 belongs to set A (is an element of set A) we write 1∈A.

To show the total number of elements that are in a given set, say set A, we use the symbol n(A). Using our example A = {1, 2, 3, 4}, then, the total number of elements of set A is 4. Symbolically , we write n(A) = 4

Description of a Set

A Set

Define a set

We describe sets either by using words, by listing or by Formula. For example if set A is a set of even numbers, we can describe it as follows:
  1. By using words: A = {even numbers}
  2. By listing: A = {2, 4, 6, 8, 10,…}
  3. By Formula: A = {x: x = 2n, where n = 1,2,3,…} and is read as A is a set of all x such that x is an even number.

Example 1

Describe the following set by listing: N is a set of Natural numbers between 0 and 11
Solution

N = {1,2,3,4,5,6,7,8,9,10}

Example 2

Write the following named set using the formula: O is a set of Odd numbers:
Solution
O = {x: x = 2n – 1, whereby n = 1,2,3….}
Example 3
Write the following set in words: W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
Solution

W = {whole numbers} or W is a set of whole numbers.

The Members of a Set

List the members of a set

The objects in a set are called the members of the set or the elements of the set.
A set should satisfy the following:
  1. The members of the set should be distinct. (not be repeated)
  2. The members of the set should be well-defined. (well-explained)

Example 4

In question 1 to 3 list the elements of the named sets.

  1. A={x: x is an odd number <10}
  2. B={days of the week which begin with letter S}
  3. C={prime numbers less than 13}

Solution

1. A={1,3,5,7,9}
2. B={Saturday,Sunday}
3. C={2,3,5,7,11}
Naming a Set
Name a set
To describe a small set, we list its members between curly brackets {, }:
  • {2, 4, 6, 8}
  • { England, France, Iran, Singapore, New Zealand }
  • { David Beckham } {}
  • (the empty set, also written ∅)

We write a ∈ X to express that a is a member of the set X. For example 4 ∈ {2, 4, 6, 8}. a /∈ X means a is not a member of X.

Differentiate Sets by Listing and by Stating the Members

Distinguish sets by listing and by stating the members

By Stating the members: A = {even numbers}

By listing: A = {2, 4, 6, 8, 10,…}

Representing Sets by using Venn Diagrams

Represent a sets by using vein diagrams

The diagrams are oval shaped. They we named after John venn, an English Mathematician who introduced them. For example A = {1,2,3} in venn diagram can be represented as follows:

is a universal set which can be a set of counting numbers and A is a subset of it.

If we have two sets, say Set A and B and these sets have some elements in common and we are supposed to represent them in venn diagrams, their ovals will overlap. For example if A = {a,b,c,d,e,} and B = {a,e,i,o,u} in venn diagrams they will look like this:

If the two sets have no elements in common, then the ovals will be separate. For example; if A = {1,2,3} and B = {5,6}. In venn diagram they will appear like here below:
If we have two sets, A and B and set A is a subset of set B then the oval for set A will be inside the oval of set B. for example; if A = {b,c} and B = {a,b,c,d} then in venn diagram it will look like this:

If we have to represent the union or intersection of two or more sets using venn diagrams, the appearance of the venn diagrams will depend on whether the sets under consideration have some elements in common or not.

Information from Venn Diagrams

Interpret information from venn diagrams
Case 1: sets with elements in common.
Example 1: If A = {5,6,7,9,10} and B = {3,4,7,9,11} represent A union B and A intersection B in venn diagrams.
Solution

Case 1: A union B

Case 2: A intersection B

Example 2: A = {a,b,c,d,e,f}, B = {a,e} and C = {b,c,e,d}. Represent in venn diagrams A∪B∪C and A∩B ∩C.

Solution: case 1. A∪B∪C

Case 2: A ∩B∩C

Case 2: sets with no elements in common:

For example; A = {a,b,c,}, B = {d,f}, C = {h,g} on venn diagram will appear like this:

Number of elements in two sets say set A and B i.e. n(A∪B) is given by: n(A∪B) = n(A) + n(B) – n(A ∩B)

proof: consider the vein diagram below:

From our vein diagram:

n(A) = x + y, n(B) = y + z, n(A∩B) = y and n(A∪B) = x + y + z thus;

n(A) + n(B) = (x + y) + (y + z)
= (x + y + z) + y
but x + y + z = n(A ∪B) and n(A∩B) = y
so,
n(A) + n(B) = n(A∪B) + n(A∩B)
make n(A∪B) be the subject of the formula
For example; if n(A) = 15, n(A∩B) = 3 and n(A∪B) = 24. Find n(B)
Soln;
Recall that: n(A∪B) = n(A) + n(B) – n(A∩B)
n(B) = n(A∪B) + n(A∩B) – n(A)
= 24 + 3 – 15
n(B) = 12

Therefore, n(B) = 12

Word problems

For example; at Mtakuja primary school there are 180 pupils. If 120 pupils like one of the sports, either netball or football and 50 pupils likes netball while 30 pupils likes both netball and football. How many pupils
  1. likes football.
  2. Likes neither of the sport
Solutions.

Let be the universal set

N be the set of pupils who likes netball
F be the set of pupils who likes football
Thus,
n(F) = ?
n(N∩F) = 30
n(N∪F) = 120
n(μ) = 180
But we know that n(N∪F) = n(N) + n(F) – n(N∩F)
Thus, n(F) = n(N∪F) + n(N∩F) – n(N)
= 120 + 30 – 50

n(F) = 100

Therefore there are 100 pupils who likes football.

2. We have a total of 180 pupils at Mtakuja primary school

But only 120 pupils likes one of either the sport. so, those who likes neither of the sport will be 180 – 120 = 50
Therefore 50 pupils likes neither of the sport.

Alternatively: by using venn diagram

n(F) only = 120 – 30 -20 = 70

n(F) = those who likes both netball and football + those who likes football only
n(F) = 30 + 70 = 100

Therefore, there 100 pupils who likes football.

Exercise 1

1. If A = {Red,White,Blue} show by using symbol that Red, White and Blue are members of set A.
2. List the elements of set B if B is a set of counting numbers.
3. Which of the following sets are finite, infinite or empty sets.
  1. A = {y:y is an odd number}
  2. B = {1,3,7,…35}
  3. C = { }
  4. D = {Maths,Biology,Physics,Chemistry}
  5. E = {Prime numbers between 31 and 37}
  6. F = {….-2,-1,0,1,2,…}
4. If A = {1,4,9,16,25,36}, B = {1,4,9} and C = {1,3,4}, which of the following statement is true:
  1. A⊂B
  2. B⊂A
  3. A⊆C
  4. C⊆B

5. How many subsets are there in set A = {f,g,I,k,m,n}? List them all.

6.If A = {all letters of English Alphabets} and B = {c,d,g,h}.List the elements of B′.
7. Let B be a set of whole numbers and C a set of prime numbers found in a set of whole numbers, using venn diagram show B∩C.

8. Draw a venn Diagram and show by shading the required region:

9. If n(A) = 90, n(B) = 120 and n(A∩B) = 45. Find:
  1. n(A∪ B)
  2. n(B) only.
  3. n(A) only.

10. In a certain meeting 40 people drank juice, 25 drank soda and 20 drank both juice and soda. How many people were in the meeting, assuming that each person took juice or soda?

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