ADDITIONAL MATHEMATICS FORM TWO NOTESN

Chapter One: Algebra

Introduction

The relationship between different quantities can easily be understood when symbols or alphabetical letters are used. For instance, if the price of an item and the number of items to be purchased are known, then the amount of money to be spent on the items can be determined. The use of symbols or letters helps in formulating suitable simultaneous equations which describe the relationship among the quantities involved. The resulting equations can be solved to determine the value of every variable. The representation of quantities and numbers in symbols or letters is known as algebra. In this chapter, you will learn how to solve simultaneous equations. The competencies developed will help you in solving different problems in business and financial management, cooking, sports, economics, engineering, health and fitness, among many other applications.

Simultaneous Equations

Simultaneous equations is a set of two or more equations each contains two or more variables whose values simultaneously satisfy each of the equations in the set. They are called simultaneous equations because their solution is obtained at the same time.

Algebraic Solutions

The solution of simultaneous equations involving linear and quadratic equations can be obtained algebraically by substitution method. In this case, one variable from one of the given equations is expressed in terms of the other and substituted into the unused equation. This results into one equation with only one variable which can be easily solved. The value obtained is substituted in one of the original equations to get the value of the second variable.

Activity 1.1: Solving simultaneous equations by substitution method

Individually or in a group, perform the following tasks:

1. Write two variable simultaneous equations of your choice involving a linear and a quadratic equation.
2. From the linear equation in task 1, make one of the variables the subject.
3. Substitute the value of the variable obtained in task 2 into the quadratic equation in task 1.
4. Solve the quadratic equation in task 3 to obtain the value(s) of the other variable.
5. Substitute the value(s) obtained in task 4 into the equation obtained in task 2 to obtain the value(s) of the variable in task 2.
6. Substitute the value(s) obtained in task 4 and 5 into the system of equations you obtained in task 1 to verify the solution.
7. What conclusion can you draw from task 6?
8. Identify activities that you think involves the concept of simultaneous equations in your everyday life.
9. Share the results with other students through discussion for more inputs.

Example 1.1

Solve the following simultaneous equations:

x + y = 5

x² - y² = 5

Solution

Let the two equations be defined as follows:

x + y = 5 ...... (i)

x² - y² = 5 ...... (ii)

From equation (i) make x the subject of the formula to obtain,

x = 5 - y ...... (iii)

Substituting equation (iii) into equation (ii) gives,

(5 - y)² - y² = 5 ...... (iv)

Expand the first term of equation (iv) to obtain,

25 - 10y + y² - y² = 5

⇒ 25 - 10y = 5

⇒ 10y = 20

⇒ y = 2

Substituting y = 2 into equation (ii) gives

x² - 2² = 5

Rearrange the equation to get,

x² = 9

x = ±3

⇒ x = 3 and y = 2 or x = -3 and y = 2

But, the set of values x = -3 and y = 2 does not satisfy both equations.

Therefore, x = 3 and y = 2.

Example 1.2

Solve the following simultaneous equations:

y = x² + 1

y - x = 1

Solution

Let the two equations be defined as follows:

y = x² + 1 ...... (i)

y - x = 1 ...... (ii)

Substituting equation (i) into equation (ii) gives,

x² + 1 - x = 1

⇒ x² - x + 1 - 1 = 0

This implies that,

x² - x = 0

Factorize the quadratic equation to obtain,

x(x - 1) = 0

⇒ either x = 0 or x - 1 = 0

Thus, x = 0 or x = 1

Substituting x = 0 into equation (i) gives,

y = 0² + 1

⇒ y = 1

Substituting x = 1 into equation (i) gives,

y = 1² + 1

⇒ y = 2

Thus, x = 0, y = 1 or x = 1, y = 2.

Therefore, the solutions are x = 0, y = 1 and x = 1, y = 2.

Exercise 1.1

Solve each of the following simultaneous equations:

1. x² + 8y = 13 and 2y + x = 2
2. 3x² - 14x - 5 = y and y = 4x - 32
3. x² + xy + y² = 28 and y = 2 - x
4. y = x² - 2x + 1 and y = ½x
5. x² + y² + 2x = 17 and y - x = 1
6. xy = 12 and y = x - 1
7. x² + y² - xy = 7 and y + x = 1
8. x² + y² = 17 and y = x + 5
9. x² + xy = 18 and 3x - y = 6
10. x² - 4xy + 4y² = 4 and 3x + 2y = 10
11. x² + 3x - y = 0 and x - y + 3 = 0
12. x - y + 5 = 0 and x² + y² - 25 = 0

Graphical Solutions

Simultaneous equations involving one linear and one quadratic equation have one or two pairs of solutions. The solutions can be obtained graphically by plotting the graphs of linear and quadratic equations on the same xy-plane. The points of intersection of the two graphs give the solutions of the two equations.

Activity 1.2: Solving simultaneous equations graphically

Individually or in a group, perform the following tasks:

1. Write two variable simultaneous equations of your choice involving a linear and a quadratic equation.
2. Prepare a table of values for each equation in task 1.
3. Use the tables of values obtained in task 2 to draw the graphs of the equations in task 1 on the same xy-plane.
4. Locate and write the points of intersection of the graphs in task 3.
5. Verify if the points obtained in task 4 are the solutions of the equations in task 1.
6. Share your results with other students through discussion for more inputs.

From Activity 1.2, it can be observed that the points of intersection of the two graphs are the solutions to the given linear and quadratic equations.

Example 1.4

Use graphical method to find the value of x and y in the following system of simultaneous equations:

y = x² - 3x + 2

y = x - 1

Solution

First prepare a table of values for the given equations for few values of x as shown in the following table:

x	-2	-1	0	1	2	3	4	5
y = x² - 3x + 2	12	6	2	0	0	2	6	12
y = x - 1	-3	-2	-1	0	1	2	3	4

The graphs of y = x² - 3x + 2 and y = x - 1 are as shown in the following figure:

Upon substitution of the points (1, 0) and (3, 2) into the given equations, it can be observed that the points satisfy the given equations.

Therefore, the solutions of the given simultaneous equations are x = 1, y = 0 and x = 3, y = 2.

Exercise 1.2

1. Use graphical method to solve each of the following pair of simultaneous equations:
   1. x² - x + 2 = y and y = 3x - 1
   2. y = x² and y = x + 6
   3. x² - 3x + 4 = y and y = x - 1
2. Use graphical method to find the points of intersection of the equations y = x² - 4 and y = x - 1.
3. Draw the graphs of y = x² - 3x - 2 and y = x + 1 on the same xy-plane. Determine the points of intersection of the two graphs and use them to find the values of each of the following:
   1. 4(x + y)
   2. (2x - y)²
4. Use graphical method to find the points of intersection of the circle x² + y² = 25 and the straight line x - y = 1.
5. If y = x² and y = 2x - 1 meet at the point P(x, y), find the values of x and y using graphical method.

Word Problems Involving Simultaneous Equations

Some of everyday real life problems lead to a pair of simultaneous equations. In such cases, the problems are transformed into mathematical symbols, letters, and operation signs. The resulting simultaneous equation can be solved algebraically or graphically.

The following steps can be used when solving word problems involving simultaneous equations:

1. Assign variables to the unknown quantities.
2. Formulate the equations by expressing the conditions in the given problem in terms of the variables.
3. Solve the resulting equations simultaneously.

Example 1.6

If the sum of two numbers is seven and their product is twelve, then find the difference between the two numbers.

Solution

Let the two numbers be x and y.

The word problem can be summarized using the following equations:

x + y = 7 ...... (i)

xy = 12 ...... (ii)

Equation (i) can be rearranged to give,

y = 7 - x ...... (iii)

Substituting equation (iii) into equation (ii) gives,

x(7 - x) = 12

Expand the equation to get,

7x - x² = 12

The quadratic equation can be rearranged to obtain,

x² - 7x + 12 = 0

Solve for x by splitting the middle term of the quadratic equation, that is,

x² - 4x - 3x + 12 = 0

⇒ x(x - 4) - 3(x - 4) = 0

⇒ (x - 3)(x - 4) = 0

Thus, either x - 3 = 0 or x - 4 = 0.

This implies that, x = 3 or x = 4.

Substituting x = 3 into equation (iii) gives,

y = 7 - 3

⇒ y = 4

Also, substituting x = 4 into equation (iii) gives,

y = 7 - 4

⇒ y = 3

Thus, x = 3, y = 4 or x = 4, y = 3.

Hence, the two numbers are 3 and 4.

The difference between the two numbers is given by, 4 - 3 = 1 or 3 - 4 = -1.

Therefore, the difference between two numbers is 1 or -1.

Exercise 1.3

1. The product of two numbers is twelve. The sum of the larger number and twice the smaller number is eleven. Find the two numbers.
2. Five years ago, a father was three times as old as his son. If the product of their present ages is 525, find their present ages.
3. A mother is p years old while her son is q years old. The sum of their ages is equal to twice the difference of their ages and the product of their ages is 675. Find the age of the son.
4. If the product of two numbers is 168 and their sum is 26. Find the numbers.
5. The area of rectangular farm is 264 m² and its perimeter is 68 m. Find its length and width.
6. If the side of the first square is twice the side of a second square and the sum of their areas is 125 cm². Find the side of each square.
7. The difference of two natural numbers is 3 and the difference of their squares is 39. Find the numbers.
8. The demand and supply of maize in our village are given by the equations, pq = 100 and 20 + 3p = q, respectively, where p and q are the price and quantity, respectively. Find the equilibrium price and quantity.
9. The perimeter of a rectangular piece of paper is 60 cm and its area is 200 cm². Find the length and width of the rectangular piece of paper.
10. The base of a triangle is 5 cm less than the height and the area is 33 cm². Find the length of the base.

Chapter Summary

1. Simultaneous equations is a set of two or more equations, each containing two or more variables whose values simultaneously satisfy each of the equations in the given set.
2. The simultaneous equations involving one linear and one quadratic equation has either one or two pairs of solutions.
3. The graphical solution of simultaneous equations involving a linear and a quadratic equation is given by the points of intersection(s) of the line and curve representing the equations, respectively.

Revision Exercise 1

1. Solve each of the following simultaneous equations:
   1. y = 2x + 1 and x² + y² = 10
   2. y = x - 3 and x² + y² = 5
   3. y = 2x - 1 and x² + xy = 24
   4. y = 2x and y² - xy = 8
   5. 2x + y = 11 and xy = 15
   6. x² + y² = 17 and y = x + 5
2. Use graphical method to determine the point of intersection of the curve x² - 5x - 12 = y and the straight line y = x - 5.
3. Solve the following simultaneous equations using graphical method; x² - 2x + 1 = y and y = x + 1.
4. Use graphical method to find the solution in each of the following:
   1. y = x² and y - x = 2
   2. x² = y + 1 and y + x = 1
5. The difference between two positive numbers is 8 and their product is 105. Find the numbers.
6. A rectangle has sides as shown in the following figure:
   1. Write down a pair of simultaneous equations using the information in the rectangle.
   2. Find the values of x and y of the simultaneous equations in (a).
7. If the length of a classroom floor is 4 m more than the width and the area is 221 m², find the dimensions of the floor.
8. Find the two consecutives even numbers whose product is 80.
9. A piece of wire 56 cm long is bent to form a rectangle with an area of 171 cm². Find the dimensions of the rectangle.
10. The perimeter of the rectangular room is 10 m. If the square of its length is reduced by 7 m² a square with a side y m is formed. By letting x and y be the length and width of the room, respectively.
    1. Form two equations relating to x and y.
    2. Find the length and width of the rectangle.
11. In a School, there are 660 students in total. If there is 40% more boys than girls. Find the number of boys and girls in the school.
12. The area of a rectangular pool is 64 m². If the length the pool is 60 m more than 4 times the width. Find the perimeter of the pool.
13. The product of present ages of Asha and John is 175 more than the product of their ages 5 years ago. If Asha is 20 years older than John. What are their present ages?
14. The length of the room is 8 m greater than the width. If both the length and the width are increased by 2 m, the area is increased by 60 m². Find the dimensions of the room.
15. An adult and five kids paid a total of Tsh 15 000 to attend the football game. Tsh 10 000 000 is the result of multiplying the adult and child ticket prices. What were the individual ticket prices for an adult and a child?

Chapter Two: Variations

Introduction

Some quantities relate to one another. The relationship in which a change in one quantity results in a proportional change in the other quantity is called variation. In this chapter, you will learn about direct, inverse, and joint variations. The competencies developed will help you to relate quantities that require changes in response to other quantities. For instance, changing the quantity of food in relation to the change in the number of people; speed of a car in relation to distance covered; demand for a commodity in relation to its price; population of a town in relation to social services; number of workers in response to the change in the number of tasks to be accomplished, among many other real life applications.

Direct Variations

Some quantities in real life relate in such a way that they change together at the same rate. For instance, change in the quantity of food relates directly to the change in the number of people. This type of relationship is called direct variation. The quantities with direct variation relationship are said to be directly proportional.

Generally, if quantities A and B vary directly, then the change in quantity A causes a proportional change in quantity B. The symbol for proportionality is ∝, hence, the direct variation of two quantities A and B is written as A ∝ B.

If A ∝ B, then A = kB, where k is a proportionality constant.

Activity 2.1: Illustrating variations of two quantities

Individually or in a group, perform the following tasks:

1. Prepare a rectangular piece of paper with dimension of your choice.
2. Use a ruler to measure and record the length and width of the rectangular piece of paper constructed in task 1.
3. Find the area of the rectangular piece of paper in task 1.
4. Divide the rectangular piece of paper in task 1 into five rectangular pieces of paper of different widths having the same length.
5. Find and record the area of each rectangular piece of paper in task 4.
6. Draw the graph of width against area using the records in task 5.
7. Use the graph from task 6 to find the relationship between width and area.
8. Comment on the relationship obtained in task 7.
9. List at least five examples of quantities you have encountered in your daily life that have similar relationship to the one found in task 7.
10. Share your results with other students through discussion for more input.

Key Point: If A₁ and B₁ are first pair of values of A and B, respectively, then second pair of values A₂ and B₂ can be computed using the following relation:

A₁/B₁ = A₂/B₂ = k

Example 2.1

Given that x varies directly as y. If x is 20 when y is 45, find:

(a) The value of y when x = 100

(b) The value of x when y = 70

Solution

(a) Given that x varies directly as y. It implies that,

x ∝ y

Introduce the constant of proportionality k to obtain,

x = ky

Making k the subject of the formula gives,

k = x/y

Substitute x = 20 and y = 45 to obtain,

k = 20/45 = 4/9

When x = 100, then

4/9 = 100/y

4y = 900

y = 900/4 = 225

Therefore, the value of y is 225.

(b) When y = 70, it implies that,

4/9 = x/70

9x = 280

x = 280/9

Therefore, the value of x is 280/9.

Example 2.2

If y varies directly as the square of x, and y = 8 when x = 2, find:

(a) The values of x when y = 20

(b) The value of y when x = 1

Solution

(a) Given y varies directly as x². It implies that,

y ∝ x²

y = kx²

k = y/x²

But y = 8 when x = 2, substitute the values to obtain,

k = 8/2² = 8/4 = 2

When y = 20, then

2 = 20/x²

2x² = 20

x² = 10

x = ±√10

Therefore, the value of x = √10 or x = -√10.

(b) When x = 1, it implies that,

2 = y/1²

y = 2

Therefore, the value of y is 2.

Exercise 2.1

1. If y varies directly as x and y = 10 when x = 15, find the value of y when x = 7.
2. Given that y varies directly as three more than the square of x and that y = 24 when x = 3. Find:
   1. The equation relating x and y
   2. The value of x which corresponds to y = 168
   3. The value of y which corresponds to x = 7
3. If A varies directly as B and the value of A = 15 when the value of B is 25:
   1. Express B in terms of A
   2. Find the value of A when the value of B = 60
   3. Find the value of B when the value of A = 18
4. If the cube root of x varies directly as the square of y and x = 216 when y = 2, find:
   1. The equation connecting x and y
   2. The value of y when x = 1728
   3. The value of x when y = 4

Inverse Variations

Inverse variation is the relationship of quantities in which the increase in one quantity causes the decrease of the other quantity and vice versa. The quantities with an inverse variation relationship are said to be inversely proportional to each other.

For instance, if a person drives from town A to town B, the more the speed of the car, the less time it takes to travel between the two towns, thus speed of the car and time of travel are said to be inversely proportional.

If p has an inverse variation relationship with q, then p ∝ 1/q

The equation relating p and q is: p = k/q, where k is a constant of proportionality.

Thus, k = pq

Activity 2.2: Determining inverse variations of two quantities

In a group, perform the following tasks:

1. Find at least 20 text books.
2. One student in the group should arrange the text books in task 1 in a shelf and another student record the time taken to complete the task.
3. Repeat task 2 using two, three, and four students in a group.
4. Use the records in task 3 to draw the graph of number of students against the reciprocals of time.
5. Use the graph from task 4 to find the relationship between the number of students and time.
6. Comment on the relationship obtained in task 5.
7. List at least five examples of quantities you have encountered in daily life that have similar relationship to the one found in task 5.
8. Share your results with other students through discussion for more input.

Key Point: If p₁ and q₁ are first set of values of p and q, respectively, then second set of values p₂ and q₂ can be computed using:

p₁q₁ = p₂q₂ = k

Example 2.6

If y varies inversely as x and y = 60 when x = 1/12, find:

(a) The constant of variation

(b) The value of x when y = 1/2

(c) The value of y when x = 100

Solution

(a) Given y varies inversely as x, it implies that,

y ∝ 1/x

y = k/x

k = xy

But y = 60 when x = 1/12, substitute the values to obtain,

k = 60 × 1/12 = 5

Therefore, k = 5.

(b) When y = 1/2, then

5 = (1/2)x

x = 10

Therefore, the value of x is 10.

(c) When x = 100, then

5 = 100y

y = 5/100 = 1/20

Therefore, the value of y is 1/20.

Exercise 2.2

1. If y varies inversely as x and that the constant of variation is 2, find:
   1. The value of y when x = 6
   2. The value of x when y = 0.7
2. If the cube root of x varies inversely as the square of y and x = 27 when y = 3, find:
   1. The constant of variation
   2. The value of x when y = 7
   3. The value of y when x = 125
3. Given that y is indirect proportional to the square of x and y = 1.25 when x = 2. Find:
   1. A formula giving y in terms of x
   2. The value of y when x = 1/4
   3. The value of x when y = 0.2

Joint Variations

A joint variation is the relationship among three or more quantities in which one quantity varies directly or inversely with two or more quantities. The quantities with a joint variation relationship are said to be jointly proportional.

For instance, if a quantity A varies directly with quantities B and C, then quantity A varies directly with both the quantities B and C. That is, A ∝ BC.

If A ∝ BC, then A = kBC, where k is a constant of proportionality.

If p ∝ q/r, then p = kq/r, where k is a constant of proportionality.

Example 2.11

If h varies jointly as l and m, such that h = 10 when l = 4 and m = 5:

(a) Find the value of the constant of proportionality

(b) Express m in terms of h and l

(c) Find the value of m when l = 20 and h = 30

(d) Find the value of h when m = 25 and l = 8

Solution

(a) Given h ∝ lm

h = klm

k = h/(lm)

But h = 10, l = 4, and m = 5, substitute the values to obtain,

k = 10/(4×5) = 10/20 = 1/2

Therefore, the constant of proportionality, k is 1/2.

(b) From h = klm, make m the subject of the formula to obtain,

m = h/(kl)

Substituting k = 1/2 gives,

m = h/((1/2)l) = 2h/l

Therefore, m = 2h/l.

(c) From m = 2h/l. If h = 30 and l = 20, it implies that,

m = (2×30)/20 = 60/20 = 3

Therefore, the value of m is 3.

(d) From h = klm. If l = 8 and m = 25, it implies that,

h = (1/2)×8×25 = 4×25 = 100

Therefore, the value of h is 100.

Exercise 2.3

1. If y varies jointly as x and z such that x = 2 when z = 3 and y = 12, find the value of y when x = 7 and z = 4.
2. If y varies jointly with x and z², such that y = 2 when x = 4 and z = 2, find the value of x when y = -1/2 and z = 1/2.
3. Given that y varies jointly with x and the square root of z. If y = 2 when x = 1/8 and z = 1/4:
   1. Write the formula of y in terms of x and z
   2. Find the value of y when x = 3/8 and z = 1/9
4. If 2 tractors can plough 6 acres of land in 4 hours. How many tractors with similar ability are needed to plough 8 acres of land in 8 hours?

Chapter Summary

1. Two quantities are said to be directly proportional if a change in one quantity causes a proportional change (increase or decrease) in the other quantity.
2. Inverse variation of two quantities is a relationship where the increase of one quantity causes a proportional decrease of the other quantity and vice versa.
3. Joint variation is a relationship among three or more quantities in which one quantity varies directly or inversely with two or more quantities.

Revision Exercise 2

1. Suppose y varies directly as x and y = 0.12 when x = 0.2.
   1. Find the value of x when y = 0.30
   2. Show that the value of y when x = 0.8 is 0.48
2. A quantity p varies inversely as the square root of w and p = 1/3 when w = 27.
   1. Express p in terms of w
   2. Find the value of w when p = √12
3. If V varies jointly as h and the square of r, and V = 45π when r = 3 and h = 5. Find the value of r when V = 175π and h = 7.
4. The temperature T of water in a lake varies inversely to the water's depth D. If at the depth of 60 m the temperature of water is 20°C, find:
   1. An equation relating temperature and depth
   2. The temperature of water at 250m
5. If x varies directly as y and inversely as z, such that x = 5 when y = 3 and z = 2.
   1. Express y in terms of other variables
   2. Find z when x = 4 and y = 6

Chapter Three: Logic

The Science of Reasoning and Argument

Chapter Progress

Introduction

The term "Logic" comes from a Greek word "Logos" which means idea, thought, word, argument, account, reason, or principle. The interest in logic is not in the statements themselves, but in how the true and false statements are related to one another. It is a reasonable way of thinking and understanding something. Formally, Logic is defined as the science or art that deals with principles of reasoning, and making appropriate decisions. In this chapter, you will learn about logical statements, truth tables, and arguments. The competencies developed will help you to distinguish between valid and invalid arguments, reason logically and make proper decisions in daily life activities, construct circuit diagrams in the field of electronics, and make logical investigations on cases and judgements in the field of law, among many other applications.

Statements

A statement or proposition can be defined as a declarative sentence that can be either true or false but not both. The truth value of a statement is either true denoted by 'T' or false denoted by 'F'.

Example 3.1

Determine whether or not each of the following sentences is a statement:

(a) Please mind your own business.

(b) x - y = z

(c) The sky is blue.

(d) Would you like some biscuits?

(e) Go to your class.

(f) One student is absent.

Solution

The sentences, (c) and (f) are statements (propositions) because their truth values are either true (T) or false (F). The sentences (a), (b), (d), and (e) are not statements (propositions) because they are neither true nor false. However, if the values of x, y and z are known, then (b) becomes a statement as it can be assigned a truth value.

Activity 3.1: Identifying statements from sentences

Individually or in a group perform the following tasks:

1. Construct any eight sentences of your choice.
2. From the sentences you have constructed in task 1, identify statements.
3. What challenges have you faced in task 2?
4. Share the results you have obtained in task 2 and 3 through discussions with other students for more inputs.

Types of Statements

There are two types of statements, namely simple and compound statements.

Simple Statement: Consists of a single declarative sentence that is either true or false.

Example: "Aisha likes singing"

Compound Statement: Formed by two or more declarative sentences.

Example: "Gabriella likes Mathematics but Musa likes History"

Truth Tables

A truth table is a table which indicates the truth value of a simple or compound statement containing several simple statements. The number of rows and columns of the truth table depends on the number of simple statements in a given compound statement.

Truth Table for One Proposition (p)

p
T
F

Truth Table for Two Propositions (p and q)

p	q
T	T
T	F
F	T
F	F

Negation of a Statement

Negation is the opposite of a given statement. The negation of a statement p is written as "~p" and it is read as "negation of p".

Truth Table for ~p

p	~p
T	F
F	T

Logical Connectives

Logical connectives are symbols used to join two or more propositions in a compound statement.

Conjunction (AND) - Symbol: ∧

The conjunction of two propositions, p and q is written as "p ∧ q" which is read as "p and q". The truth value is true if and only if both p and q are true.

p	q	p ∧ q
T	T	T
T	F	F
F	T	F
F	F	F

Disjunction (OR) - Symbol: ∨

The disjunction of two propositions, p and q is written as "p ∨ q" and it is read as "p or q". The truth value is false if and only if both propositions are false.

p	q	p ∨ q
T	T	T
T	F	T
F	T	T
F	F	F

Conditional (Implication) - Symbol: →

The conditional statement "p → q" is read as "If p, then q" or "p implies q". Its truth value is false when p is true and q is false.

p	q	p → q
T	T	T
T	F	F
F	T	T
F	F	T

Biconditional (Double Implication) - Symbol: ↔

The biconditional statement "p ↔ q" is read as "p if and only if q". The truth value is true if both p and q have the same truth value.

p	q	p ↔ q
T	T	T
T	F	F
F	T	F
F	F	T

Example 3.8

Write the statement "If she plays the drums, then he plays the trumpet" in symbolic form.

Solution

Let p: She plays the drums.

Let q: He plays the trumpet.

The statement becomes "if p, then q", which is an implication statement.

Therefore, the symbolic form of the given statement is p → q.

Tautologies and Contradictions

A tautology is a compound statement which is always true for every truth value of its individual proposition.

Example: "Either Mwanaidi will go home or she will not go home."

A contradiction is a compound statement which is always false for every truth value of its individual proposition.

Example: "5 is an even number and 5 is not an even number."

Activity 3.2: Recognizing a tautology or contradiction

Individually or in a group, perform the following tasks:

1. Construct at least four compound statements of your choice.
2. Write the compound statements constructed in task 1 into symbolic form.
3. Construct a truth table for each of the symbolic statements in task 2.
4. Identify the truth value for each of the symbolic statement in task 3.
5. Use the results in task 4 to recognize compounds statements which are tautology, contradiction, or neither.
6. Share your results with other students through discussion for more inputs.

Arguments

An argument is a statement with a given set of hypotheses (premises) which yield a given conclusion.

Example 3.18

Test the validity of the argument "If I like music, then I will study. I did not study. Therefore, I don't like music".

Solution

Let p: I like music.

Let q: I will study.

The premises and conclusion of the argument are written as: p → q; ~q ∴ ~p

Thus, the argument in symbolic form is written as: [(p → q) ∧ ~q] → ~p

Using truth tables, we can verify that this argument is valid.

Activity 3.3: Recognizing validity of an argument

Individually or in a group, perform the following tasks:

1. Construct at least six simple statements of your choice.
2. From task 1, identify at least three related statements that form an argument.
3. From task 2, write down the premises and the conclusion.
4. Use the premises and the conclusion in task 3 to formulate the argument.
5. Construct the truth table for the argument formulated in task 4.
6. From task 5, identify the truth value of the argument.
7. From task 6, with reason(s), state whether the argument is valid or not.
8. Share your results with other students through discussion for more inputs.

Exercise 3.1

Determine whether each of the following statements is simple, compound or not a statement:

1. A positive integer is a prime number only if it has no divisors other than 1 and itself.
2. 3 < -5
3. Go to sleep!
4. If it rains tonight, then I will stay at home.
5. What time is it?

Chapter Summary

1. A statement or proposition can be defined as a declarative sentence that can be either true or false, but not both.
2. There are two types of statements; namely simple and compound statements.
3. The number of cases that describe the given compound statement depends on the number of simple statements in the compound statement.
4. A truth table is a table that describes the truth values of a compound statement.
5. Negation of a statement is written by introducing the word 'not' along with a verb.
6. Logical connectives include conjunction (AND), disjunction (OR), conditional (implication), and biconditional (double implication).
7. A tautology is a compound statement which is always true.
8. A contradiction is a compound statement which is always false.
9. An argument is a statement with a given set of hypotheses (premises) which yield a given conclusion.
10. If an argument is a tautology, then the argument is valid, otherwise is not valid.

Revision Exercise 3

1. Determine whether each of the following sentences is a simple statement, a compound statement, or not a proposition at all:
   1. What is your name?
   2. One year is equivalent to fifty weeks.
   3. A person is helpful if and only if he is kind.
2. Let p be the statement "12 is divisible by 3" and q be the statement "2 is even". Express each of the following proposition as an English sentence:
   1. ~q ∧ (q ∨ p)
   2. ~q ↔ (p ∧ q)
3. Use truth tables to determine whether each of the following statements is a tautology, contradiction or neither:
   1. [(p → q) → q] → p
   2. ~[(~p ↔ q) ↔ (q ↔ p)]

← Previous Chapter: Variations Next Chapter: Locus →

Chapter Four: Locus

The Path of Moving Points

Chapter Progress: Locus Concepts

Introduction

The movement of an object from one point to another can be described by its location. The path traced by a moving object which satisfies stated set of conditions is called locus. For instance, the earth revolves around the sun, the centre of the wheel of a vehicle moving on a straight road, and the lines drawn on the middle of the road each forms locus. In this chapter, you will learn about a locus about a fixed point, two fixed points, a line, and two intersecting lines. The competencies developed will help you in construction of bridges, flyovers, roads, sports and games run ways in the field of engineering, among many other applications.

Locus About a Fixed Point

A locus is defined as a path traced by a moving object. Geometrically, it is a set of all points whose location is determined by one or more specified conditions. For instance, a line, a circle, an ellipse, a parabola, and many other shapes are described by the loci of points. The plural of locus is loci and it is derived from the word "location".

Geometrical Representation

When an object moves such that, it is always at a constant distance from a fixed point, its locus is described as a circle.

Fixed Point (O): Center of the circle

Moving Point (P): Any point on the circumference

Constant Distance: Radius (OP)

Activity 4.1: Constructing a locus about a fixed point

Individually or in a group, perform the following tasks:

1. Locate a fixed point O on a graph paper, and a point P at a distance of your choice from a fixed point O to represent a moving object.
2. Put the end point of the compass at a point O and its pencil end at P.
3. Construct an arc by moving the pencil end away from point P.
4. Continue drawing an arc until the pencil end of the compass meets the point P again.
5. Describe the name of the figure obtained.
6. Repeat using different distances of point P from a fixed point O.
7. Share your results with other students through discussion for more inputs.

Example 4.1

Construct the locus of a point Q at a constant distance of 3 cm from a fixed point E.

Solution

The locus of a point Q is a circle with centre at a fixed point E and radius 3 cm.

Construction:

1. Mark point E as the center

2. Set compass to 3 cm radius

3. Draw a complete circle around E

4. All points on this circle are 3 cm from E

Cartesian Representation of Locus About a Fixed Point

Equation of a circle: (x - a)² + (y - b)² = r²

Where (a, b) is the center and r is the radius

Example 4.4

Find the equation of the locus of the point P(x, y) which moves on a plane such that its distance from the origin is always 3 units.

Solution

The locus of a moving point P(x, y) is a circle with centre at the origin and radius r units.

Equation: x² + y² = r²

Substitute r = 3: x² + y² = 9

Therefore, the equation of the locus is x² + y² = 9

Example 4.5

A point is moving on the xy-plane such that its distance from a fixed point (3, 4) is always 5 units. Find the equation of the locus.

Solution

Using the formula: (x - a)² + (y - b)² = r²

Where (a, b) = (3, 4) and r = 5

(x - 3)² + (y - 4)² = 25

Expanding: x² - 6x + 9 + y² - 8y + 16 = 25

Therefore, x² + y² - 6x - 8y = 0

Locus About Two Fixed Points

Perpendicular Bisector

The locus of a point which moves such that it is equidistant from two fixed points A and B is the perpendicular bisector of the line joining the two points.

Construction Steps

1. Locate two points A and B at a reasonable distance apart
2. Draw line segment AB connecting the points
3. Adjust compass to obtain a length more than half of AB
4. Place compass pointer at A and draw arcs above and below AB
5. Maintaining the radius, place compass pointer at B and draw arcs to cross the previous arcs
6. Mark the intersection points as X and Y
7. Join points X and Y with a straight line
8. This line XY is the perpendicular bisector

Example 4.8

Construct the locus of a point P which is equidistant from the two fixed points A and B such that AB = 8 cm.

Solution

The locus is the perpendicular bisector of the line AB.

Construction:

1. Draw line AB = 8 cm

2. Find midpoint of AB

3. Construct perpendicular line through midpoint

4. This perpendicular line is the required locus

Cartesian Representation of Locus About Two Fixed Points

Distance Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Example 4.12

A point E moves so that it is equidistant from the points M(0, 5) and N(2, 7). Show that the equation of the locus of E is x + y - 7 = 0.

Solution

Let E be the point (x, y). Since point E is equidistant from M and N:

EM = EN

√[(x - 0)² + (y - 5)²] = √[(x - 2)² + (y - 7)²]

Squaring both sides:

(x - 0)² + (y - 5)² = (x - 2)² + (y - 7)²

x² + y² - 10y + 25 = x² - 4x + 4 + y² - 14y + 49

Simplifying: 4x + 4y - 28 = 0

Therefore: x + y - 7 = 0

Locus About a Line

Parallel Lines as Locus

The locus of a point moving such that it is always equidistant from a straight line is a pair of lines parallel to the straight line.

Example 4.14

Construct the locus of a point P that moves at a constant distance of 2 cm from a straight line AB.

Solution

The locus is a pair of parallel lines 2 cm from the line AB.

Construction:

1. Draw line AB

2. Using a set square, construct lines parallel to AB at 2 cm distance on both sides

3. These parallel lines form the required locus

Locus About Two Intersecting Lines

Angle Bisectors

The locus of a point which is equidistant from two intersecting straight lines consists of a pair of straight lines which bisect the angles between the two given lines.

Example 4.19

Construct the locus of a point P such that it is always equidistant from two intersecting lines AB and CD.

Solution

The locus is the angle bisectors of the angles formed between intersecting lines AB and CD.

Construction:

1. Draw two intersecting lines AB and CD

2. Construct the angle bisectors of all four angles formed

3. These bisectors form the required locus

Exercise 4.1

1. The student radio station has a broadcasting range of 4 km. Describe the locus of points which represents the outer edge of the broadcasting range.
2. Two points R and P are moving such that they are always 2.5 cm and 4 cm, respectively from the same fixed point O. Construct the loci of the points R and P.
3. Find the equation of the locus of a point A(x, y) which moves so that it is 4 units from the origin.
4. Find the equation of the locus of a point P(x, y) which is moving so that it is always 3 units from the point (-2, -3).

Chapter Summary

1. A locus is defined as a path traced by a moving point.
2. The locus of a point moving such that it is always at constant distance from a fixed point is a circle.
3. A point moving such that it is always equidistant from two fixed points forms a perpendicular bisector of the line joining the two fixed points.
4. The distance between two fixed points P(x₁, y₁) and Q(x₂, y₂) is given by d = √[(x₂ - x₁)² + (y₂ - y₁)²].
5. The locus of a point moving such that it is always at a certain distance from a straight line is a pair of parallel lines.
6. The locus of a point moving such that it is always equidistant from two intersecting lines is the bisector of the angle between the two lines.

Revision Exercise 4

1. Construct the locus of a moving point such that it is always 5cm from a fixed point.
2. Find the equation of the locus of a point C which moves such that it is k units from the origin.
3. If a point A is at unit distance from the point (-1,2). Find the locus of the point A in Cartesian form.
4. Two buildings are 2 km apart. The canal is to be dug such that the distance from any point on the canal to each building is always the same. Describe where the canal should be dug.
5. Find the equation of the locus of a point which is equidistant from points (1,2) and (3,4).

← Previous Chapter: Logic Next Chapter: Sets →

Chapter Five: Sets

Organizing and Categorizing Objects

Chapter Progress: Set Theory Concepts

Introduction

Different objects with common characteristics can be categorized and well-studied when arranged together in groups. For instance, a group of form two students' names, a collection of coins, a group of even numbers less than ten, and so on. In this chapter, you will learn about operations on sets and number of elements in a set. The competencies developed in this chapter will help you to fulfill various tasks such as organizing, creating, categorizing objects, among many other applications.

The Concept of Sets

A set is a collection of well-defined distinct objects. It is denoted by a capital letter, with its elements enclosed within a curled brackets {} and separated by commas. The elements of a set are represented by small letters, numbers, symbols or words.

Set Notation

If a is an element of set A, then in set notation is written as a ∈ A.

If a is not the element of set A, symbolically it is written as a ∉ A.

A set containing all elements under consideration is called a universal set, it is denoted by the letter U.

Example 5.1

If U = {2, 7, 8, 9, 12, 13, 15, 16, 17, 19, 21, 29}, A = {2, 8, 13, 17, 29}, B = {7, 12, 13, 16, 17, 21}, and C = {8, 9, 12, 16, 17, 19}, find:

(a) A ∪ B ∪ C

(b) A ∩ B ∩ C

(c) B' ∪ A

Solution

(a) A ∪ B = {2, 8, 13, 17, 29} ∪ {7, 12, 13, 16, 17, 21} = {2, 7, 8, 12, 13, 16, 17, 21, 29}

A ∪ B ∪ C = {2, 7, 8, 12, 13, 16, 17, 21, 29} ∪ {8, 9, 12, 16, 17, 19} = {2, 7, 8, 9, 12, 13, 16, 17, 19, 21, 29}

(b) A ∩ B = {2, 8, 13, 17, 29} ∩ {7, 12, 13, 16, 17, 21} = {13, 17}

A ∩ B ∩ C = {13, 17} ∩ {8, 9, 12, 16, 17, 19} = {17}

(c) B' = {2, 8, 9, 15, 19, 29}

B' ∪ A = {2, 8, 9, 15, 19, 29} ∪ {2, 8, 13, 17, 29} = {2, 8, 9, 13, 15, 17, 19, 29}

Activity 5.1: Collecting together items of the same characteristics

Individually or in groups, perform the following tasks:

1. Using manila card or any other card of your choice, prepare one hundred pieces of cards with convenient dimensions.
2. Using marker pens or pencils, label the pieces of cards prepared in task 1 with numbers from 1 to 20 in five pairs.
3. From task 2, prepare a set A whose elements are even numbers less than 10 and odd numbers greater than 10.
4. From task 2, prepare a set B of odd numbers less than 17.
5. From task 2, prepare a set C of prime numbers.
6. From task 2, form a set D whose elements are either in sets A, B, or C without repeating the elements.
7. From task 2, form a set E of common elements found in sets A, B, and C without repeating the elements.
8. List the elements that do not belong to a set B.
9. Suggest the names of the sets operations used in tasks 6, 7, and 8.
10. Identify various sets that you have encountered in your daily life.
11. Are there any sets identified in task 10 with common elements? If so list them.
12. Share your results with other students through discussions for more inputs.

Operations on Sets

Union of Sets

The union of two or more sets is the set that contains all the elements that are in all the sets without repetition. The union is denoted by the cup symbol "∪".

If x ∈ (A ∪ B ∪ C), then x ∈ A, x ∈ B or x ∈ C.

Intersection of Sets

The intersection of two or more sets is the set that contains all the elements that are common to all the sets. The intersection is denoted by the cap symbol "∩".

If x ∈ (A ∩ B ∩ C), then x ∈ A, x ∈ B, and x ∈ C.

Complement of a Set

The complement of a set A is the set that contains all the elements that are in the universal set but not in the set A. It is denoted by A' or A^c.

If x ∈ A', then x ∉ A.

Venn Diagram Representation

Venn diagrams are used to show the logical relationship between sets.

Number of Elements in a Set

Cardinality of Sets

The number of elements of a set is a count of individual elements in a set. It is also known as the cardinality of the set and is denoted by n(A), reads "the number of elements in the set A".

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ C) - n(B ∩ C) + n(A ∩ B ∩ C)

Example 5.4

If n(U) = 68, n(A) = 41, n(C) = 21, n(B) = 17, n(A ∩ B) = 19, n(A ∩ C) = 23, n(B ∩ C) = 11, and n(A ∩ B ∩ C) = 27, find:

(a) n(A ∪ B ∪ C)

(b) n(A ∪ B ∪ C)'

Solution

(a) Using the formula:

n(A ∪ B ∪ C) = 41 + 17 + 21 - 19 - 23 - 11 + 27

= 106 - 53 = 53

(b) n(A ∪ B ∪ C)' = n(U) - n(A ∪ B ∪ C)

= 68 - 53 = 15

Example 5.6

A secondary school carried out a monthly test and the results were as follow. Out of 1,350 candidates, 600 passed Geography, 700 passed History, 350 passed English Language and 50 failed all the three subjects. Also, 200 passed Geography and History, 150 passed Geography and English Language, 100 passed History and English Language.

(a) How many candidates passed in all the three subjects?

(b) Illustrate the given information in a Venn diagram.

Solution

Let G, H, and E represent sets of candidates who passed Geography, History, and English respectively.

n(U) = 1350, n(G) = 600, n(H) = 700, n(E) = 350

n(G ∪ H ∪ E)' = 50, n(G ∩ H) = 200, n(G ∩ E) = 150, n(H ∩ E) = 100

n(G ∪ H ∪ E) = 1350 - 50 = 1300

Using the formula:

1300 = 600 + 700 + 350 - 200 - 150 - 100 + n(G ∩ H ∩ E)

1300 = 1200 + n(G ∩ H ∩ E)

n(G ∩ H ∩ E) = 100

Therefore, 100 candidates passed all three subjects.

Practical Applications

Real-life Set Applications

Sets are used in various real-life situations:

• Organizing student data in schools
• Categorizing products in supermarkets
• Classifying books in libraries
• Grouping employees in organizations
• Sorting data in computer systems

Example 5.8

An investigator was paid 1,000 Tanzanian shillings per person interviewed about like and dislike on food served for lunch. He reported that 200 like rice, 190 like chips, 260 like Ugali, 60 like chips and rice, 40 like chips and ugali, 100 like rice and ugali, 30 like all three types of food, while 105 people do not like any food at all.

(a) Summarize this information in a Venn diagram.

(b) Hence, calculate the amount of money paid to the investigator.

Solution

Let R, C, and M represent sets of people who like rice, chips and ugali respectively.

Using Venn diagram principles and the inclusion-exclusion principle, we can calculate the total number of people interviewed.

Total people = Sum of all regions in the Venn diagram

Amount paid = Total people × 1000 Tsh

Exercise 5.1

1. Given that U = {a, b, c, d, f, g, h, j, k, w, l, m, n, t}, A = {a, d, f, g, h, j, k}, B = {g, h, j, k, l, m, n}, and C = {a, b, c, d, f, g, h, j}
   1. Present the given sets in a Venn diagram.
   2. Shade the region represented by (C' ∩ A') ∪ (B' ∪ C)'.
   3. Shade the region not represented by (A ∪ B') ∩ (B ∪ C').
2. If U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {2, 3, 4} and B = {5, 6, 7}
   1. Find: (i) A' ∪ B' (ii) A' ∩ B'
   2. Show that (A ∪ B)' = A' ∩ B'.

Exercise 5.2

1. Given three sets, A, B, and C such that n(A) = 70, n(B) = 62, n(C) = 59, n(A ∩ B) = 25, n(A ∩ C) = 15, n(B ∩ C) = 19, and n(A ∩ B ∩ C) = 12. Find:
   1. n(A ∪ B ∪ C)
   2. n(A ∩ B')
   3. n(B ∩ C')
   4. n(A' ∩ C)
2. Each student in a Form two class of 40 students plays at least one of the following games; volleyball, football, and tennis. There are 18 students who play volleyball, 20 students who play football, and 27 students who play tennis. If 7 students play volleyball and football, 12 students play football and tennis, and 4 students play all games. Use formulas to calculate the number of students who play:
   1. Volleyball and tennis.
   2. Volleyball and tennis but not football.
   3. Just one game.
   4. Football or tennis.

Chapter Summary

1. Set is a well-defined collection of distinct objects.
2. Union of sets is a set that contains all elements of all sets.
3. Intersection of sets is a set that contains all elements that are common in all sets.
4. Complement of a set is the set that contain all elements of the universal set that are not in the given set.
5. Universal set is a set which contains all the elements under consideration.
6. Venn diagram is the diagram used to show the logical relationship between sets.
7. Number of elements in a set is the simple count of individual elements in a set.
8. The formula for three sets: n(A∪B∪C) = n(A) + n(B) + n(C) - n(A∩B) - n(A∩C) - n(B∩C) + n(A∩B∩C)

Revision Exercise 5

1. If A, B, and C are subsets of the universal set U, then shade the region represented by each of the following:
   1. (A ∩ B') ∪ (A ∪ B)'
   2. A ∩ (B ∪ C)'
   3. (A ∩ C) ∪ (C' ∪ A)'
2. If U = {105, 115, 125, 135, 145, 155, 165, 175, 185}, D = {115, 125, 135, 145, 155}, E = {105, 115, 145, 165}, and F = {115, 145, 165, 175}, then find:
   1. (D ∪ E') ∩ (E ∪ F')
   2. (D ∪ E ∪ F)'
   3. (D ∩ E) ∪ (E ∩ F)
3. Given that U = {red, black, pink, yellow, blue, white, magenta, purple}, A = {red, black, pink}, B = {black, yellow, blue}, and C = {black, white, magenta} Find:
   1. (A' ∪ B')' ∪ (A' ∪ B)'
   2. A' ∪ (C ∩ B)'
   3. B ∪ (A ∩ C)

← Previous Chapter: Locus Next: Additional Resources →