Topic 1: Mappings and Relations

By the end of this topic, you will be able to:

(a) use arrow diagrams/mappings to represent relations and functions.

(b) identify the domain and range of mapping.

(c) describe and distinguish between function and non-function mapping.

Keywords

• domain
• function
• many-to-one mapping mapping
• non-function
• one-to-one mapping range
• relation

Introduction

Mapping and relations are very important because they develop a connection between related items, objects or numbers. In this topic, you will be able to understand and use arrow diagrams/ mappings to represent relations and functions.

What is a relation?

Activity 1.1 Discovering what relation is

Look at the pictures provided. Use them to answer the questions below.

1. Identify and name the different objects in the picture.

2. How would you connect the different items you have identified in (1)?

3. Is there any connection between the ball, chicken, eggs and the chicks? If yes/no give an explanation to your answer.

4. Share your work with your classmates.

1.1 Using arrow diagrams to represent relation and mapping

Activity 1.2 Identifying relations between items

1. In Mathematics, you can find statements such as:
   • “2 is a prime number among whole numbers”.
   • “5 is less than 8”.
   • “24 is the lowest common multiple of 6 and 8”.
     • In each of the statements, identify
       • (a) the items being connected.
       • (b) the statement that connects each item identified in (a).
2. Here are some English statements you always meet:
   • “Guundi is a friend of Kanya”.
   • “Ntuza is taller than Nyanin”.
   • “Musa comes from the same country as Haruna”.
   • “Mukwano is a brother of Nyako”.
     • In each of the statements, identify
       • (a) the items being connected.
       • (b) the statement that connects each item identified in (a).

Research corner

Look around your environment. Identify situations that explain the meaning of a relation.

(a) Prepare a PowerPoint presentation that you will share with your classmates.

(b) Where you can, use a camera to take pictures of the situations you have identified.

(c) If possible, you may role-play the statements and illustrate your findings.

Exercise 1.1

1. Juliana, a daughter to Mariam, is a niece to Sarah.

(a) Identify all the relations in the statement with a connecting statement made.

(b) Illustrate the relations by use of arrows, clearly stating the connecting statement.

2. Given the set D = (-3, -2, -1, +1, +2, +3) and set E = {1, 4, 9) (a) Draw a diagram to illustrate the relation is “a square of” between set D and E. (b) What other relations can you obtain between set D and set E?

3. Two students illustrated the relation is a square of as follows. One presented it as 2 >>>4 and another presented it as 4>>>> 2. Who of the two students is correct? Give an explanation of your result.

4. Write down the set A of prime numbers less than 13.

(a) What is the relationship between set A and set B = {4, 6, 8,9,15, 22}?

(b) Suggest a way of connecting the relation between set A and set B using a diagram.

5. Draw an arrow diagram to show the relations defined by the following sets.

(a) A= {2, 3, 4, 5, 6, 7} B={10, 12, 14, 15, 18): is “a factor of”

(b) D={6, 8, 9, 10, 11) E=(9, 11, 12, 13, 14, 15): is “three less than”

Learning points

1. Relations clearly define the connection between one set and another set.
2. A relation maps the element of one set called the object to another set called the image. The elements in the two sets are connected by an arrow.

1.2 Identifying domain and range of a mapping

In the previous section, you should have observed that arrows are used to show a relation between sets. When you used an arrow to connect two sets, then you obtained an arrow diagram.

Activity 1.3 (a) Identifying domain and range

Study the diagram below and use it to answer the questions that follow.

1. From the diagram:

(a) name set A and set B. B 2.

(b) what is the relation that relates set A and set B?

(c) For the relation you have chosen in (b), identify the object and the image set.

2. How else would you use the relation identified in (1)? Illustrate it diagrammatically with the aid of the arrows.

Activity 1.3 (b) Finding out more about relations

1. You are provided with cards, each having digits 2, 3, 5, 7, 6, 15, 18, 20, 35.

(a) Arrange them by forming two sets so that each digit is associated with its prime factors.

(b) Identify the members in the object set and the image set.

2. (a) Draw a diagram to connect the two sets in (1) using the relation “is a prime factor of”.

(b) What is the object set and the image set?

Learning point

The first set is the object which is referred to as the domain and the second set is the image which is also referred to as the range.

Exercise 1.2

1. The following are pairs of coordinates: {(0,1), (1, 3), (2, -4), (-4,1)). Write the domain and the range.
2. P = {2, 3, 5, 7} and Q = = {4, 9, 49, 25}
   • (a) State the relationship between P and Q.
   • (b) Construct an arrow diagram between P and Q.
   • (c) If 8 was a member of P, what is the image of 8 in Q?
3. The relation 3t+ 4, maps 2 to 10.
   • (a) Explain how 2 has been mapped onto 10.
   • (b) Find the elements of the range when the domain is P = {1, 2, 3, 4, 5}. (c) Find the elements of the domain when the range is Q = {22, 25, 28, 46, 49}.
4. “Is a prime factor” is a relation from A to B indicated in the mapping diagram below.
   • (a) What are the elements of the domain indicated in the diagram?
   • (b) Write the elements of the range in the diagram
   • (c) If B was the domain, what relation would you use to obtain elements of the range?
   • (d) Draw an arrow diagram to illustrate the relation in (c).
5. H is the relation “is a factor of” between the sets A = {2, 3, 4, 5, 7} and B= (6, 12, 30,35). With the aid of a diagram, illustrate the relationship between set A and set B.
6. Let D=(-2, -1,0 1, 2) and E = {1, 0, 4).
   • (a) Draw an arrow diagram to show the relation is “the square root of”.
   • (b) Write down:
     • (i)the members of the domain.
     • (ii) the members of the range.

1.3 Describing and distinguishing between function and non-function mapping

So far you have learnt to distinguish a mapping from a relation and use the words range and domain. You have also learnt how to illustrate a relation using an arrow diagram by mapping two sets. In this section, you will learn different types of mappings and how they can be used to differentiate between a function and a non-function.

Activity 1.4 Identifying the types of mappings Look at the arrow diagram C.

Look at the arrow diagram C.

(a) What is the relation that maps A to B?

(b) List the elements of the domain and range.

(c) How many elements are in each of the sets of the domain and range?

(d) How many arrows map each element of the domain to each element of the range?

(e) If in set B another city was added. The city is Gulu. Would it qualify to be mapped using the relation in

(f)? Are there elements in set A having more than one element mapped onto set B? (g)

What type of mapping best describes the arrow diagram?

2. Look at the arrow diagram D.

(a) What relation maps A onto B?(b) Is it possible with the relation you have stated in (a) to have 2 elements in the domain being mapped onto the same element in the range? Give a reason for your answer.(c) What difference do you observe about the arrow diagrams C and D?

(d) What type of mapping best describes the arrow diagram in D?

3. Look at the arrow diagram E.

(a) List all elements in the domain and range.

(b) What relation maps each element of set A to each element of set B in arrow diagram E?

(c) What do you notice about arrow diagrams C, D and E?

(d) What type of mapping is represented in arrow diagram E?

(e) Does this mean the same? Black Uganda and Uganda – Explain your response. Black

Learning points

• A function is a mapping in which every element in the domain is mapped onto one and only one element in the range A.
• One-to-one and many-to-one mappings are functions. Therefore arrow diagram C describes a function whereas D and E are non-functions.
• A many-to-many mapping is a non-function.

1. Identify any areas in real life that describe a one-to-one mapping. many-to-many mapping, many-to-one mapping and one-to-many 2.

2. mapping In each case draw a picture that illustrates the situation that describes the different types of mappings identified in (1).

Exercise 1.3

1. Let A and B denote the sets of animals and their young ones respectively. A=(cat, dog, cow, goat). B(kitten, puppy, calf, kid).
   • (a) Draw an arrow diagram that connects A and B.
   • (b) What type of mapping is represented by set A and set B?
2. Given the elements of the domain as T = {1, 2, 3, 4,…, 28, 29, 30):
   • (a) Use the relation “is four times” to find all the elements of the range.
   • (b) Draw an arrow diagram to illustrate the relation. 3
3. Is this mapping a function and how can you tell? 2 5 4.

1. Is this set of ordered pairs a function and how can you tell? (-4, 6), (2,-4), (3, 8), (3,-12) and (5,1) 5.
2. In the given relation, what domain value corresponds to the range value-2? ((-1, 2), (2, 4), (2, 5), (0,-2), (2, 0))
3. Mappings and Relations Look at the pairs of coordinates given in (a) to (d). Which of the following is not a function?
   • (a) {(0, 1), (1, 2), (2, 3), (3, 4)}
   • (b) {(0, 2), (1, 3), (4, 3), (1, 2)}
   • (c) {(1, 3), (4, 2), (2, 0), (3, 4)}
   • (d) {(1, 2), (2, 2), (3, 2), (4, 2)}
4. Which of the arrow diagrams given below represents a function? Give reasons to support your answer.