MATHEMATICS As LEVEL(FORM FIVE) NOTES – ET THEORY

ET THEORY

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

• Sets are denoted by capital letters e.g. A, B, C, D etc.
• The statement “x is an element of A” or “x belongs to A” is written as x ∈ A.
• If x is not an element of A, we write x A.

Important sets of the number system

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

IR⁺: a set of positive real numbers.

IR^–: 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, numbers that repeat and terminate).

N: a set of natural numbers (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. Describe its elements by mathematical notation or actual words (set builder notation).

Examples

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

Solution: A is the set of all prime numbers less than 15.

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

Solution: Since x² = 9, x = 3 or x = -3.

B =

The general form of set builder notation:

A =

OR

A =

Example: 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

A =

b) B =

B =

c) C =

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:

K =

Y =

A =

b) B =

y =

B =

BASIC CONCEPTS OF SET

1. The set that does not contain any element is called an empty set, denoted 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 the same elements.
4. Equivalent: two sets are equivalent if they have the same number of elements.
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 = then A ⊆ B.
7. Proper subset: Suppose A = and B = then A ⊂ B.

Note:

• (i) Φ (an empty set) is a subset of any set.
• (ii) A ⊆ A (a set is a subset of itself).

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 B =

Subsets of A are: Subsets of B are:

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

If a set has n members, the number of subsets = 2ⁿ.

THE POWER SET

Is a set which contains all subsets of the given set.

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 A ∪ B.

• A ∪ B =
• Is a set which has elements of set A or set B without repetition.

Examples

→ If A = and B =

A ∪ B =

→ If A = and B =

A ∪ B =

2. INTERSECTION

Is a set which has elements contained in both 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 the universal set but not in A.

Example: 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 =

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:
   1. A =
   2. 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 ∈ Φ
   • d) Φ ∈ Φ
4. Let A =
   1. Is ∈ A?
   2. Is 2 ∈ A?
   3. Is ∈ A?
   4. Is ∈ A?
   5. Is ∈ A?
   6. Is ∈ A?
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?
   1. A ⊆ B
   2. A ⊂ B
   3. A ∩ B
   4. A′
   5. B′
   6. A B

QUESTIONS

1. Let µ be the universal set and Φ be an empty set. What are:
   1. Φ = µ
   2. µ = Φ
   3. µ – Φ = µ
   4. Φ – µ = Φ
   5. µ ∩ Φ = Φ
   6. µ ⊆ Φ = µ
2. Let A be subset of the universal set µ. What are the following?
   1. A ⊆ Φ = A
   2. A ⊆ A = A
   3. A ⊆ Φ = Φ
   4. A ⊆ A = A
   5. A ⊆ µ = A
   6. A ⊆ µ = µ
   7. A ∩ A′ = Φ or {}
   8. A ⊆ A′ = µ
   9. A ∩ µ = A′
   10. A ⊆ Φ = A
3. Let A and B be subsets of a universal set µ. Suppose A ⊆ B. What are:
   1. A ∪ B = B
   2. A ⊆ B = A

SET INTERVAL ON THE NUMBER LINE

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

Solutions:

For A =

For B =

Examples

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

a) A ⊆ B

b) A′

c) B′

d) A ∪ B′

Solutions:

a) A ⊆ B

A ⊆ B =

b) A′

A′ =

c) B′

B′ =

d) A ∪ B′

A ∪ B′ =

QUESTIONS

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

(ii)(b) A ∪ B

(ii)(c) B′

Uses of Venn diagram

• To illustrate set identities.
• To find the number of members in a given set.

1. Illustration of set identity

Example: Illustrate by use of Venn diagram (A ∪ B) ∩ A = A.

Solution:

Two different methods can be used:

1. Shading method
2. Numbering of disjoint subsets

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

Shade (A ∪ B) ∩ A by vertical lines and A by horizontal lines.

Now (A ∪ B) ∩ A = region shaded = A = R.H.S.

∴ (A ∪ B) ∩ A = A.

ii) Numbering of disjoint subsets

Solutions:

L.H.S = (A ∪ B) ∩ A

Now A ∪ B = subsets 1, 2, 3

But A = subsets 1, 2

(A ∪ B) ∩ A = A = R.H.S.

Example:

Use Venn diagram to show A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

Solution:

L.H.S = A ∩ (B ∪ C)

Now B ∪ C = subsets 5, 6

A ∩ (B ∪ C) = subsets 1, 2, 5, 4 and 6

R.H.S = (A ∩ B) ∪ (A ∩ C)

A ∩ B = subsets 1, 2

A ∩ C = subsets 1, 2, 3, 4, 5, 6, 7

(A ∩ B) ∪ (A ∩ C) = 1, 2, 5, 4, 6

QUESTIONS

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 ∪ B = B ∪ A

A ∩ B = B ∩ A

2. Associative laws

a) (A ∪ B) ∪ C = A ∪ (B ∪ C)

b) (A ∩ B) ∩ C = A ∩ (B ∩ C)

3. Distributive laws

a) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

b) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

4. De Morgan’s laws

a) (A ∪ B)′ = A′ ∩ B′

b) (A ∩ B)′ = A′ ∪ B′

5. Identity laws

a) A ∪ µ = µ

b) A ∩ µ = A

c) A ∪ Φ = A

d) A ∩ Φ = Φ

e) AΦ = A

f) AA = 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′) ∩ (A ∪ B′))′ Associative law

≡ (µ ∩ (A ∪ B′))′ Complement law

≡ (µ)′ Identity law

≡ A Complement law

≡ (A ∪ B) Identity law

Exercise

1. Use laws of algebra of set to simplify:

i) (A ∪ (A ∩ B′))

ii) (A′ ∩ B′) ∪ (A ∩ B)

iii) (A ∩ B) ∪ (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: x² – x – 2 = 0}. Find n(A).

Solution:

n(A) = 2

Note:

• The number of elements of a set is defined only for a finite set.
• If A ⊆ µ then the number of elements of A′ is n(A′) = n(µ) – n(A).

Example

If A ⊆ µ and B ⊆ µ 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 subsets as follows:

Let n(A ∩ B′) = a, n(A′ ∩ B) = c, n(A ∩ B) = b.

n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

= (a + b) + (b + c) – b

= a + 2b + c – b

= a + b + c

Examples

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:

1. n(A)
2. n(A ∩ B′)
3. n(µ)
4. n(A′ ∩ B)

Solutions

1. n(X ∪ Y) = n(X) + n(Y) – n(X ∩ Y) = 18 + 26 – 12 = 32

2. n(S ∩ T) = n(S) – n(S ∩ T′) = 15 – 10 = 5

3. i) n(A) = n(A – B) + n(A ∩ B) = 8 + 5 = 13

ii) n(A ∩ B′) = n(A) + n(B′) = 13 + 4 = 17

iii) n(A ∩ B′) = 17

iv) n(µ) = n(A ∩ B) + n((A ∩ B)′) = 16 + 4 = 20

n(µ) = 20

iv) n(A′ ∩ B) = n(B) – n(A ∩ B) = 16 – 13 = 3

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:

1. How many students study math only?
2. How many students study science only?
3. How many students study both math and science?

2. In a class of 35 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 – 3 + 4 – 4 + 4 = 15

18 – 3 = 15

= 3

ii) Number of animals eating grass = 6 + 5 + 2 + 3 = 16 animals.