Topic 13: Set Theory

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

(a) describe a set and identify elements of a set.

(b) identify different types of sets and their symbols (empty set, universal set, complement set, disjoint set, intersection set, union set, subset).

(c) determine the number of elements in a set.

(d) represent and show different operations on sets by shading the different regions in a Venn diagram.

(e) apply sets in practical situations using two and three sets.

Keywords

• complement
• disjoint
• element, member of equal sets
• equivalent sets
• intersection
• null, empty set
• set
• subset
• union
• universal set

Introduction

Set theory is an important topic for every human being. Shopkeepers, market vendors, manufacturers and even agriculturalists use sets in their everyday work in one way or another. For example, check how the market vendors organize their items on stands. Are fruits placed with fish? Why not? The study of this topic will thus help you be able to use sets to solve problems.

13.1 Describing a set and identifying elements of a set

Activity 13.1 Describing a set

A market vendor wants to organise her produce. She sells watermelon, mangoes, onions, garlic, avocadoes, apples, pineapples, green pepper and carrots.

1. Group her produce in terms of fruits and vegetables. 2. Share your ideas with your classmates.

Examples

1. Identify the elements of set K = {the even numbers less than 15).
2. Write down the set L of elements which are odd numbers less than 15.
3. Identify the set M which consists of the multiples of 3 less than 15.
4. Write down the elements of set N which is a set of square numbers between 1 and 15. 5.
5. Describe set P = {4, 6, 8, 10, 12}.

Solution

1. K = {2, 4, 6, 8, 10, 12, 14}

2. L = {1, 3, 5, 7, 9, 11, 13}

3. M = {3, 6, 9, 12}

4. N = {4, 9}

5. P is a set of even numbers between 3 and 13.

Learning points

Sets contain members with a similarity. It is easy to identify a set once the similarity is given.

Exercise 13.1

Describe the following sets.

(a) A= {2, 4, 6, 8, 10, 12, 14, …}

(b) B={1, 3, 5, 7, 9, 11, 13, 15, …}

(c) C={1, 4, 9, 16, 25, 36, 49, 64, …}

(d) D={1, 3, 6, 10, 15, 21, 28, 36, 45, …}

(e) E = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}

(f) F = {January, February, March, April, May, June, …, December}

(g) G={January, March, May, July, August, October, December}

(h) H = {April, June, September, November} (i) M = {3, 6, 9, 12, 15, 18, 21, 24, …,}

(i) T={1, 2, 3, 6, 9, 18, …}

(k) W = {…, -3, -2, -1, 0, 1, 2, 3,…}

13.2 Identifying different types of sets and their symbols

In primary school, you studied about different types of sets and how they are represented. If two sets P and Q are such that all members of Q are also members of P, how do you describe set Q? In this section, you will describe different sets using various set symbols such as:U, n, p, &, E etc.

Activity 13.2 Identifying set symbols

Set A has prime numbers between 0 and 10. Set B has even numbers between 0 and 10.

1. List elements in set A and set B.

2. How many members are in each set?

3. How many elements are in both set A and set B and which elements are these?

4. How many elements are in both set A and set B all together?

Learning points •

Number of members in a set is given by n(set).

Elements shared by sets are the intersection i.e. AnB and a combination of all members in the provided sets is given by AUB.

Exercise 13.2

1.Bag A contains bottles, plates and cups. Bag B contains spoons, forks, salt and sugar.

(a) Represent the two sets on a Venn diagram.

(b) Find set AnB. (c) What is the general name for the two sets A and B?

2. Given that A = {bottles, plates, cups}, list down all the subsets of set A.

13.3 Determining the number of members in a set

Activity 13.3 Determining the number of members in a set

You are provided with two boxes :A and B.

1. List the elements in box A and box B.

2. How many elements are in box A? You can write this as n(A)=?

3. Carefully study the members in sets A and B listed in (1). How many elements are in set:

(a) B?

(b) ANB?

(c) AUB?

Exercise 13.3

1. In a village, 64 farmers planted maize (M) and 42 farmers planted bananas (B) while 32 farmers planted other crops (C) but not maize or bananas. The farmers who grew maize or bananas are 86. Of the farmers who planted maize, some planted both maize and bananas.

(a) Find the number of farmers who planted:

(i) both maize and bananas.

(ii) maize only.

(ii) bananas only.

(b) Find the total number of farmers in the village.

(c) State

(i) n(M) (ii) (iv) n(MOB) (v) n(B) n(MNC) (iii) n(C) (vi) n(BNC)

2. In a class, 15 learners like studying Mathematics only, 18 like studying English only and 12 like studying both Mathematics and English. Given that 25 like studying other subjects, find:

(a) the number of learners who like studying

(i) Mathematics.

(ii) English.

(b) how many learners are in the class.

13.4 Representing and showing different operations on sets by shading in a Venn diagram

In your primary school, you studied Venn diagrams and represented sets on Venn diagrams. You will extend this idea in this section and be able to identify certain regions on a well-drawn Venn diagram by shading.

Activity 13.4 Drawing Venn diagrams

1. Given the sets A = {pen, pencil, paper, book, mug, spoon) and B= {table, chair, book, mug, spoon}:

(a) Draw a Venn diagram to represent the sets.

(b) Find

(i) n(ANB).

(ii) n(AUB).

(c) Re-draw the diagram in (a) above and shade out regions representing

(i) ANB

(ii) AUB

(iii) A’

(iv) ANB’

2. Present your work to the rest of the class.

Exercise 13.4

1. Given the universal set & = {1, 2, 3, 7, 8, 9} where A = {2, 4, 6, 8} B = {2, 3, 5, 7}

(a) Draw a Venn diagram to show the above information.

(b) Shade out AnB.

(c) Draw another diagram and shade the following sets:

(i) ANB’

(ii) A’NB

(iii) AUB’

(iv) A’NB’

(v) A’UB’

2. Given that, N= numbers}.

(a) Set Theory (first five even numbers) and M = {first five odd List the members in the two sets N and M.

(b) Represent the two sets on a Venn diagram.

(c) What name do you give to the two sets?

13.5 Applying sets in practical situations

There are a number of world problems that can best be solved by sets. Whenever you go to market to buy, say, clothes, there are certain properties you will always want to see on the clothes. Do you find it easy to get an article of clothing with all the needed properties? Most likely no. In such a case, you look for an article of clothing with more desirable properties than all the others. In this section, you will apply sets to solve practical problems.

Activity 13.5 Solving practical situations using sets

1. You are provided with a bag containing a number of items. In your group, use the available packaging materials to sort the items properly. Explain why you have sorted them the way you have done and why it is important to sort items.
2. Carry out a survey among your members regarding what foods they would prefer, given matooke (M), rice (R) and posho (P). Carry out the same survey two other groups in your class. Merge your findings and represent the information on a Venn diagram. Of what importance is such a survey?
3. Present to the rest of the class.

Exercise 13.5 1.

1. The Ministry of Public Service advertised for suitably qualified personnel to fill vacancies of Chief Administrative Officer (CAO) existing in newly created districts in Uganda. Of the 93 who were shortlisted for interview, 53 had experience in a related field (E), 52 had a master’s degree (M); 57 had no criminal record (C); 25 belong to both E and C, 27 to both M and C while 31 to both E and M.
   • (a) Represent this information in a Venn diagram, clearly showing the number of applicants in each region.
   • (b) Use the Venn diagram to find the number of applicants who had
     • (i) all the three qualifications.
     • (ii) exactly two of the qualifications.
     • (ii) only one of the qualifications.
   • (c) Giving reasons, who among these applicants would recommend for the job of CAO? you
2. At a certain school, there are 76 learners in S.6 who study science subjects: 32 study Physics, 40 study Chemistry and 48 study Biology; 20 of them study Biology and Chemistry, 14 study Physics and Chemistry while 16 study Physics and Biology. On the other hand,6 study all the three subjects.
   • (a) Represent the information clearly on a Venn diagram.
   • (b) Find the number of learners who study at least two of the science subjects.
   • (c) Determine the number of learners who study Chemistry and Physics only.
   • (d) How many learners study Biology only?
3. In a certain hotel, 39 guests ordered either matooke, cassava or rice. 24 ordered matooke (M), 16 ordered rice (R) and 17 ordered cassava (C). Those who ordered matooke and cassava were more than those who ordered both cassava and rice by one person; 9 ordered for matooke and rice; 2 ordered all the three types of food.
   • (a) Represent the information on a Venn diagram.
   • (b) Find how many guests ordered both cassava and rice.