SET THEORY

SET THEORY

The word set is used to denote a collection of well defined objects

Set are denoted by capital letters e.g. A, B, C, D etc

The statement ‘’ x is an element of A’’ or ‘’ x belong to A’’ is written as x ∈ A

If x is not an element of A, we write x A

**Importance sets of the number system**

IR: a set of real numbers (+, -) all numbers

IR^{+}: Is a set of positive real numbers

IR^{–}: Is a set of negative real numbers

Z: a set of integers. (+, -) whole numbers

Z^{+}: a set of positive integers

Z^{–}: a set of negative integers

Q: a set of rational numbers (rational ½ = 0.33333 – rational numbers, number repeats and terminate

N: a set of natural number (positive numbers starting from 1, 2, 3…… counting numbers)

**SPECIFICATION OF A SET**

There are two ways of specifying a set;

1. List its members (roster method)

2. Describing its elements by mathematical notation or actual words (builder notation).

Examples

1. Let A = specified in roster form, specify this by set builder notation

**Solution**

A is a set of all prime numbers less than 15

2. Let B = specified by set builder, specify by roster form

**Solution**

Since x^{2} = 9, x = 3, x = -3

B =

The general form of set builder notation

A =

OR

A =

E.g. A =

**QUESTIONS**

1. Let A =

a) Is 10 ∈ A NO

b) Is 11 ∈ A NO

c) Is 13 ∈ A NO

d) List all elements of A

A =

2. Use the roster method to specify the following sets

a) A = {x ∈ Z: x + 3 = 5}

x + 3 = 5; x = 5 – 3, x = 2

x = -0.5 and x = 0.5

C=

3. Specify the following in roster form

a) A = {y ∈ Z: y= 3K where K∈Z^{+} and K ≤ 6}

Solution

**BASIC CONCEPTS OF SET.**

1. The set that does not contain any element is called an empty set, donated by Φ or { }

2. Universal set is a set which contains all elements under consideration. It is denoted by µ.

3. Equality; two sets are equal if they have same elements

i.e. If A = and B =

4. Equivalent; two sets are equivalent if they have same number of elements

i. e A = and B = ∴A≡B

5. Subsets; A is a subset of B if every member of A is also a member of B. It is denoted by A B

6. Improper subset; suppose A = and B = A B

7. Proper subset. Suppose A = and B = A B

Note i) (an empty set is subset of any set)

ii) A A (a set is subset of its own set)

**Number of subsets in a set**

Let S =

How many subsets does it have?

The subsets are: { }

→There are 8 subsets of S.

If A = and If B =

Subset of A are : Subsets of B are :

Number of subsets of A= 2 Number of subset of B = 4

If a set has n members, the number of subsets = 2^{n}

**THE POWER SET**

Is a set which contains all subsets of the given sets

If A = , subsets are

Power set of A is given by S =

Given B =

The power set of B is given by

S =

**OPERATION OF SETS**

** 1. UNION**

The union of two sets A and B is denoted by AUB

– AUB =

– Is a set which have elements of set A or set B without repetition.

**Examples**

→If A = and B =

AUB =

→If A = and B =

AUB =

** 2. INTERSECTION**

– Is a set which have both elements contained in set A and set B

A∩B = {x:x∈A and x∈B}

Examples

→If A = and B =

A B =

→If A = and B =

A B =

Here A and B are disjoint sets.

** 3. COMPLEMENT**

The complement of Set A denoted by A′ is the set of all elements which are in universal set but not in A.

E.g. A =

µ=

A′ =

** 4. RELATIVE COMPLEMENT**

Relative complement of A with respect to set B is denoted by A’ B or A – B and is defined as follows

A B =

Example

A =

B =

Then A B =

B A =

** 5. THE SYMMETRIC DIFFERENCE**

All elements which are either in set A or set B but not both

– The symmetric difference of A and B is denoted by A B

A B =

**Examples**

A =

B =

A B =

**QUESTIONS**

1. List the subsets of the following sets

a) A =

b) B =

2. Let A =

Write down the subsets of A

3. Which of the following are true and which are false?

a) Φ Φ b) 0 = Φ c) Φ∈ d) Φ ∈

4 . Let A =

a) Is ∈ A

b) Is 2 ∈ A

c) Is ∈ A

d) Is A

e) Is

f) Is

5. Let µ be the set of all positive integers, A is the set of all even integers and B is a set of all odd integers. What are sets?

a) A B b) A B c) A B d) A’ e) B’ f) A B

** QUESTIONS**

1. Let µ be the universal set and Φ be an empty set. What are

a) Φ = µ

b) µ = Φ

c) µ – Φ = µ

d) Φ – µ = Φ

e) µ ∩ Φ = Φ

f) µ Φ = µ

2. Let A be subset of the universal set µ. What are the following?

a) A Φ = A

b) A A = A

c) A Φ = Φ

d) A A = A

e) A µ = A

f) A µ = µ

g) A ∩ A’ = Φ or {}

h) A A^{‘}= µ

i) A µ = A^{‘}

j) A Φ = A

3. Let A and B be subsets of a universal set µ. Suppose A B. What are;

a) A U B = B

b) A B = A

**SET INTERVAL ON THE NUMBER LINE**

1. Let A = and B={x∈IR:-7< x ≤ 3}Represents these set intervals on two separate number lines

Solutions

For A =

For B =

**Examples**

Using the sets A and B defined above, state and represents the following sets on same number line

a) A B b) A′ c) B′ d) A U B′

Solutions

a) A B

A B =

b) A′

A′ =

c)B′

B′ =

a)

(d)A U B′

A U B′ =

**QUESTION**

i) Represent the above sets on one number line

ii) Draw and state each of the following sets on separate number lines

a) A ∩ B b) A ∪B c) B′ d) A∩B′

Solution

(i)

(ii)(a) A

b) A U B

c) B′

**QUESTIONS.**

1. Represents and then draw on one number line the following set interval

Using the above set interval, represent and state the following

i) A B ii) A C iii) C B^{‘} iv) (A B) C

**VENN DIAGRAMS**

Sets can be represented in the form of diagrams called Venn diagrams

– The universal set is represented by a rectangle

– Subsets of U are represented by a circle in universal set** **

**Uses of Venn diagram**

i) To illustrate sets identity

ii) To find number of members in a given set

1. Illustration of set identity

Example; Illustrate by use of Venn diagram (A U B) A = A

Solution.

Two different methods can be used

i) Shading method

ii) Numbering of disjoint subsets

i) Shading method, i.e. to show (A B) ∩ A = A

L. H. S → (A B) ∩ A

Shade (A B) by vertical lines

Shade (A B) A by horizontal lines

Now (A B) A = region shaded

= A

= R. H. S

∴ (A B) A = A

** ii) Numbering of disjoint**

Solutions

L. H. S = (A B) A

Now A B = subsets 1, 2, 3

But A = sub 1, 2

(A B) A = subsets 1, 2

=A

= R. H. S

**Example**

Use Venn diagram to show A (B C) = (A B) (A C)

Solution

L. H. S = A U (B C)

Now B C subsets 5, 6

A U (B C) Subsets 1, 2, 5, 4 and 6

R. H. S = (A U B) (A U C)

A U B subsets 1, 2, 3, 4, 5, 6

A U C subsets 1, 2, 3, 4, 5, 6, 7

(A U B) ∩ (A U C) = 1, 2, 5, 4, 6

A (B C) = (AB) (A C)

**QUESTION**

Use a Venn diagram to show the following

i) (A B) A = A

ii) A (B C) = (A B) (A C)

**LAWS OF ALGEBRA OF SETS**

Set operations obey the following laws

**1**.** Commutative laws**

A U B = B U A

A B = B A

**2. Associative laws**

a) (A U B) U C = A U (B U C)

b) (A B) C = A (B C)

**3. Distributive laws**

a) A U (B C) = (A U B) (A U C)

b) A (B U C) = (A B) (A C)

**4. De -Morgan’s laws**

a) (A U B)′ = A′ B′

b) (A B)′ = A′U B′

**5. Identity laws**

a) A µ = µ

b) A µ = A

c) A Φ = A

d) A Φ =Φ

e) A\Φ = A

f) A\A = Φ

**Examples**

Use laws of algebra of set to simplify

1. (A (A B)′)′

Solution

(A (A B)′)′ ≡(A (A′ B′))′ De-Morgan’s law

≡((A A′) B′ )′Associative law

≡ (Φ B′) Complement law

≡ (Φ)′Identity law

≡ µ complement law

(A(A U B)′)′ = µ

**Examples**

Use the laws of algebra of sets to prove

(A (B C′)) C = (A C) (B C)

Solution

L.H.S (A (B C′)) C

= (((A B) C′) C…….. Associative law

=((A B) U C) (C′ C) ………distributive law

= ((A B) C) (µ) …………complement law

= (A B) C……………. identity law

= (A C) (B C) ……………distributive law

= R. H. S

**Exercise**

1. Use laws of algebra of set to simply

i) (A B) (A B’)

ii) (A’ B’) (A B)

iii) (A B) U (A – B)

iv) A (A B)

2. Use laws of algebra to prove

i) (Z W)′ W = Φ

ii) (X Y’) (X Y) (Y X′) = X Y

iii) (A – B) A = A

Note

A – B = A B′ by definition

**Number of elements in a set**

The number of elements in set A is denoted by n (A)

**Example**

Let A be a set of all positive odd integers which are less than 10. Find n (A)

Solution

A = {1, 3, 5, 7, 9}

Now n (A) = 5

**Examples**

Let A ={x ∈ IR:x^{2}-x-2=0}. Find n (A)

**Solution**

**Note**

i) The number of elements of a set is defined only for a finite set

ii) If A U then the number of elements of A′ is n(A′) = n(µ) – n(A)

Example

If A U and B U then show that n (A B) = n(A) + n(B) – n(A B)

Proof

Refer to the Venn diagram below

Represents the number of elements in disjoint subset as follows

Let n (A B′) = a n (A′ B) = c

n (A B) = b

R. H. S = n (A) + n (B) – n (A B)

= (a + b) + (b + c) – b

= a + 2b + c – b

= a + b + c

n (A B)

L. H. S

**EXAMPLE**

1. Given n (X) = 18, n (Y) = 26, n (X ∩ Y) = 12. Find n (X Y)

2. Given n (S T) = 19, n (s) = 15. n (S T′) = 10. Find n(S T)

3. Given n (A B) = 15

n (A B) = 16

n ((A B)′) = 4

n (A – B) = 8

Find i) n (A) ii) n (A B′) iii) n (µ) iv) n (A′ B)

**Solutions**

1. n (X Y) = n (X) + n (Y) – n (X Y)

=18 + 26 – 12

= 32

n (X Y) = 32

2. n(S T) = 19, n(S) = 15, n(S T’) = 10

i) n(S T) =?

n(S T) = n(S) – n(S T’)

= 15 – 10

= 5

n( S T) = 5

3. n(A B) = 5, n(A B) = 16, n(A B)′ = 4, n(A – B) = 8

i) n(A) =? ii) n (A B′) iii) n(µ) iv) n (A′B)

**Solutions**

n (A) = n(A – B) + n(A B)

= 8 + 5

= 13

ii) n (A B’) = n (A) + n(B′)

= 13 + 4

= 17

n(A B’) = 17

iii) n(µ) = n(A B) + n(( A B))′

= 16 + 4

= 20

n(µ) = 20

iv) n(A′ B) = n(B) – n(A B)

n(A′ B) = n(A B) – n(A) + n(A B) – n (n (A B))

n(A′ B) = 16 – 13

n (A′ B) = 3

4. By using n (A B) = n(A) + n(B) – n(A B) show that;

n(A B C) = n(A) + n(B) + n(C) – n(A B) – n(A C) – n(B C) + n(A B C)

**Solutions**

Let B C = K

L.H.S n(A B C) = n(A K)

= n(A) + n(K) – n(A K)

= n(A) + n(B C) – n(A (B C))

= n(A) + n(B) + n(C) – n(B C) – n((A B) (A C))

= n(A) + n(B) + n(C) – n(B C) – (n(A B) + n( A C) – n((A B)(A C))

= n(A) + n(B) + n(C) – n(B C) – n(A B) – n(A C) + n(A B C)

**Questions**

There are 26 animals in zoo, 5 animals eat all type of food in the zoo i.e. grass, meat and bones. 6 animals eat grass and meat only, 2 animals eat grass and bones only, 4 animals eat meat and bones only. The number of animals eating one type of food only is divided equally between the three types of food.

i) Illustrate the above information by a labeled Venn diagram

ii) Find the number of animals eating grass

**Solutions
**

Let M set of animals that eat meat

Let B set of animals that eat bones

Let G set of animals that eat grass

3 + 6 + 5 + 4 + 2 = 26

3 + 17 = 26

= 3

ii) Number of animals eating grass

= 6 + 5 + 2 + 3

= 16 animals

**Questions **

1. A class has 15 boys and 15 girls. In the class 20 students are studying science, 14 students are studying math, 10 boys are studying science, 10 boys are studying math, 8 boys are studying both math and science, 4 girls are studying neither math nor science.

Find i) How many students study math only?

ii) How many students study science only?

iii) How many students study both math and science?

2. In a class of 35 students each students each student takes either one of two subjects (physics, chemistry and biology). If 13 students take chemistry, 22 students take physics,17 students take biology, 6 students take both physics and chemistry and 3 students take both biology and chemistry. Find the number of students who take both biology and physics.

**Solutions**

Since there are 15 girls

10 – + + 4 – + 4 = 15

18 – = 15

= 3

i) Students who study math only = 2 + 1

= 3 students** **

** **ii) Students who study science only = 2 + 7

= 9 students

iii) Students who study both math and science = 8 + 3

= 11 students

**QUESTIONS**

1. In a certain college apart from other discipline, no students is allowed to study less than two of the subjects, finance, accounting and economics, 150 students study finance, 110 study accounting, 80 study economics and 20 study three subjects

i) How many students study two of the named subjects?

ii) How many study finance or accounting or economic?

2. One poultry farm in Dar produces three types of chicks and in six months report revealed that out of 126 of its regular customers, 65 bought broilers, 80 bought layers and 75 bought cocks, 45 bought layers and cocks, 35 bought broilers and cocks, 10 bought broilers only, 15 bought layers only and bought cocks only, 6 of the customers did not show up.

i) How many customers bought all the three products?

ii) How many customers bought exactly two of the products?

3. An investigator was paid sh. 100 per person interviewed about their likes and dislikes on a drink for lunch. He reported 252 responded positively coffee, 210 liked tea, 300 liked soda, 80 liked tea and soda, 60 liked coffee and soda. 50 liked all three, while 120 people said they not like any drink at all. How much should the investigator be paid coffee and tea 60 people?

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