Subject :

Mathematics

Topic :

Logarithms

Class :

SS 1

Term:

First Term

Week:

Week 7

Instructional Materials:

• Wall charts
• Online Resources
• Pictures
• Related Audio Visual
• Mathematics Textbooks

Reference Materials

• Scheme of Work
• Online Information
• Textbooks
• Workbooks
• Education Curriculum

Previous Knowledge :

The pupils have previous knowledge of

Logarithms

Behavioural Objectives: At the end of the lesson, the students should be able to

• define set
• mention types of set
• solve questions on set

Content:

WEEK 7

DATE……………………….

Subject:    Mathematics Class:       SS 1

TOPIC:   SETS

Content:

• Definition of sets
• Set notations
• Types of sets

SUB-TOPIC 1: Definition

Definition of sets

A set is a well-defined list

A set is a collection of distinct objects, considered as an object in its own right. For example, the set of all planets in the solar system is a proper subset of the set of all bodies in the solar system; since Jupiter is a planet, it therefore belongs to that subset, but not to the entirety of the solar system.

In mathematics, a set is usually denoted by a capital letter. The objects that make up a set are called its elements, and the notation x ∈ S (“x is an element of S”) is used to denote that x is an element of the set S. If x is not an element of S, then x ∉ S (“x is not an element of S”) is used. Sets are conventionally denoted by capital letters.

The empty set, often denoted by ∅ or { } , is the unique set with no elements; it is also called the null set. The empty set is a subset of every other set, including itself.

The union of two sets A and B, denoted by A ∪ B , is the set of all elements that belong to either A or B (or both). The intersection of A and B , denoted by A ∩ B , is the set of all elements that belong to both A and B .

Examples:

1. The set of all natural numbers (N)

2. The set of all real numbers (R)

3. The set of all complex numbers (C)

4. The set of all prime numbers (P)

5. The set of all irrational numbers (I)

or collection of objects with some characteristics which are unique to its members.

Examples:

• a set of mathematics text books
• a set of cutleries
• a set of drawing materials

Sometimes there may be no obvious connection between the members of a set. Example:

{chair, 3, car, orange, book, boy, stone}.

Each item in a given set are normally referred to as a member or element of the set.

SUB-TOPIC 2: SET NOTATION

This is a way of representing a set using any of the following.

• Listing method
• Rule method or word description
• Set builders
• Listing Method

A set is usually denoted by capital letters and the elements in it can be defined either by    making a list of its members. Eg A = {2, 3, 5, 7}, B = {a, b, c, d, e, f, g, h, i} etc.

Note that the elements of a set are normally separated by commas and enclosed in curly brackets or braces

• Rule Method . The elements in a set can be defined also by describing the rule or property that connects its Eg C = {even number between 7 and 15. D=

{set of numbers divisible by 5 between 1 and 52.}, B = {x : x is the factors of 24}etc

• Set–Builders Notations

A set can also be specified using the set – builder notation. Set – builder notation is an algebraic way of representing sets using a mixture of word, letters , numbers and inequality symbols e.g. B = {x : 6 ≤ x < 11, x є ƶ} or B = {x/6 ≤ x < 11, x є ƶ}. The expression above is interpreted as “B is a set of values x such that 6 is less than or equal to x and x is less than 11, where x is an integer (z)”

• The stroke (/) or colon (:) can be used interchangeably to mean “such that”
• The letter Z or I if used represents integer or whole

Hence, the elements of the set A = {x : 6 ≤ x < 11, x є ƶ} are A = { 6, 7, 8, 9,10}.

NB:

• The values of x starts at 6 because 6 ≤ x
• The values ends at 10 because x < 11 and 10 is the first integer less than

The set builder’s notation could be an equation, which has to be solved to obtain the elements of the set. It could also be an inequality, which also has to be solved to get the range of values that forms the set.

EVALUATION

• Define Set

(a) C = {x : 3x – 4 = 1, x є ƶ}

• P = {x : x is the prime factor of the LCM of 15 and 24}

© Q = {The set of alphabets}

• R = {x : x ≥ 5, x is an odd number}

Types of sets

1. There are many types of sets in mathematics, but the most common ones are finite sets, infinite sets, and subsets.

2. A finite set is a set that has a definite, finite number of elements. An example of a finite set is the set of all whole numbers from 1 to 10

3. An infinite set is a set that has an infinite number of elements. An example of an infinite set is the set of all whole numbers

4. A subset is a set that is contained within another set. An example of a subset is the set of all even numbers which is contained within the set of all whole numbers

5. The empty set is a set that has no elements. An example of an empty set is the set of all whole numbers that are less than 1.

Examples 1:

List the elements of the following sets

(i)         A = {x : 2 < x ≤ 7, x є ƶ}.

• B = {x : x > 4, x є ƶ}
• C = {x : -3 ≤ x ≤ 18, x є ƶ}.

(iv)       D = {x : 5x -3 = 2x + 12, x є Z}.

(v)        E = {x : 3x -2 = x + 3, x є I}

(vi)       F = {x : 6x -5 ≥ 8x + 7, x є ƶ}

• P = {x : 15 ≤ x < 25, x are numbers divisible by 3}
• Q = {x : x is a factor of 18, }

Solution:

(i)         A = {3, 4, 5, 6, 7}

Note that:

– the values of x start at 3, because 2 < x

-The values of x ends at 7 because x ≤ 7 i.e. because of the equality sign. (ii)      B = {5, 6, 7, 8, 9, .. .}

Note that:

the values of x start from 5 because 5 is the first number greater than 4 (i.e. we are told that x is       greater than 4)

(iii)       C = {-3, -2, -1, 0, 1, . . , 15, 16, 17, 18}

Note that:

–           The values of x starts from -3 because -3 ≤ x, and ends at 18 because x ≤ 18 (there is equality sign at both                           ends).

• To be able to list the elements of thisset, the equation defined has to be solved

i.e. 5x – 3 = 2x + 12

5x – 2x = 12 + 3

3x = 15

x     =¹⁵/₃

∴ x      =      5

∴ D    = {5}

• We also need to solve the equation to get the set values 3x – 2 = x + 3

3x – x  =  3 + 2

∴ x = ⁵/₂

2x     =  5

Since ⁵/₂ is not an integer (whole number) therefore the set will contain no element.

∴є       = { } or Ø

• Solving the inequality to get the range of values for the set, we have 6x – 5 ≥ 8x + 7

6x – 8x ≥ 7 + 5

-2x ≥ 12 x ≤ ¹²/_-2

∴x ≤ -6

∴F = {…, -8, -7, -6}

(vii)      P = {15, 18, 21, 24}

Note that:

The values of x start at 15 because it is the first number divisible by 3 and fallswithin the range defined.

(viii)     Q = {1, 2, 3, 6, 9, 18}

Example 3:

Rewrite the following using set builder notation

(i)         A = {8, 9, 10, 11, 12, 13, 14}

(ii)        B = {3, 4, 5, 6 . . . }

(iii)       C = {. . . 21, 22, 23, 24}

(iv)       D = {7, 9, 11, 13, 15, 17 . . .}

(v)        P = {1, -2}

(vi)       Q = {a, e, i, o, u}

Solution:

• A = {x : 7 < x < 15, x є ƶ} OR A = {x : 8 ≤ x < 15, x є ƶ} OR A = {x : 7 < x ≤ 14, x є ƶ} OR A = {x : 8 ≤ x ≤ 14, x є ƶ}
• B = {x : x > 2, x є ƶ} OR B = {x : x ≥ 3, x є ƶ}
• C = {x : x < 25, x є ƶ} OR C = {x : x ≤ 24, x є ƶ}
• D= {x : x> 8 or x≥ 7, x is odd, xєƶ}
• P={1,2} suggests the solutions of a quadratic Therefore , the equation or set- builders notation can be obtained from :

x² – (sum of roots)x + product of roots = 0 x² –(-1)x + (1 x -2) = 0

x² + x – 2 = 0

P = {x : x² + x – 2 = 0, x є ƶ}

• Q = {x : x is a vowel}

EVALUATION

1. List the elements in the following Sets

(a)         A = {x : -2 ≤ x < 4, x є ƶ} (b) B = {x : 9 < x < 24, x є N}

(c)        C = {x : 7 < x ≤ 20, x is a prime number, x є I} (d)    D = {x / 2x – 1 = 10, x є Z}

(e)        P = {x : x are the prime factor of the LCM of 60 and 42}

2. Rewrite the following using Set – builder (a) Q = {. . . 2, 3, 4, 5}

(b)        A = {2, 5}

(c)        B = {2, 4, 6, 8, 10, 12 . . .}

(d)        A = {-2, -1, 0, 1, 2, 3, 4, 5, 6}

(e)        C = {1, 3, -2}

SUB-TOPIC 4: TYPES OF SETS

Finite Sets

Refers to any set, in which it is possible to count all the elements that make up the set.

These types of sets have end. E.g. A = {1, 2, 3,  . . , 8, 9, 10}

B = {18, 19, 20, 21, 22}

C = {Prime number between 1 and 15} etc.

Infinite    Sets

Refers to any set, in which it is impossible to count all the elements that make up the set. In other words, members or elements of these types of set have no end. These types of set, when listed are usually terminated with three dots or three dots before the starting values showing that the values continue in the order listed. E.g.

(i)         A = {1, 2, 3, 4, . . }

(ii)        B = {…,-4,-3,-2,-1,0,1,2,3,…}

(iii)       C= {Real numbers} etc.

Empty or null Set

A set is said to be empty if it contains no element. Eg {the set of whole number that lies between 1 and 2}, {the set of goats that can read and write}, etc Empty sets are usually represented using ø or { }.

It should be noted that {0} is NOT an empty set because it contains the element 0, Another name for empty set is null set.

Number   of Elements    in a Set

Given a set A = {-2, -1, 0, 1, 2, 3, 4, 6} the number of elements in the set A denoted by n(A) is 9;  i.e. n (A) = 7

If B = {2,3, 5} then n (B) = 3

If Q = {0} then n (Q) = 1

Other examples are as follows:

Example 4:

Find the number of elements in the set: P = {x : 3x -5 < x + 1 < 2x + 3, x є ƶ }

Solution:

3x – 5 < x + 1 and x + 1 < 2x + 3 3x – x < 1 + 5 and x – 2x < 3 – 1 2x < 6      – x < 2

x <⁶/₂                       x > -2

x < 3                      -2 < x

-2 < x < 3

The integers that form the solution set are P =  {-1, 0, 1, 2}

∴ n {P} = 4

Example 5:

Find the number of elements in the set A = {x : 7 < x < 11, x is a prime number}

Solution:

The set A = { } or Ø since 8, 9, 10 are no prime numbers.     ∴ n(A) = 0

Example 6:

Find the number of elements in the following sets:

• B = {x : x ≤ 7, x є ƶ}
• C = {x : 3 < x ≤ 8, x is a number divisible by 2}.

Solution:

(i)         B = {. . . 3, 4, 5, 6, 7}.

The values of the set B has no end     hence it is an infinite set i.e. n(B) = ∞

(ii)        C = {4, 6, 8}.   ∴ n(C) = 3

The Universal    Set

This is the Set that contains all the elements that are used in a given problem. Universal Sets vary from problem to problem. It is usually denoted using the symbolsξor μ.

Note that when the Universal Set of a given problem is defined, all values outside the universal set cannot be considered i.e. they are invalid.

EVALUATION

(1) List the elements in the following Sets

(a)         A = {x : -2 ≤ x < 4, x є ƶ} (b) B = {x : 9 < x < 24, x є N}

(c)        C = {x : 7 < x ≤ 20, x is a prime number, x є I} (d)    D = {x / 2x – 1 = 10, x є Z}

(e)        P = {x : x are the prime factor of the LCM of 60 and 42} (2). Find the number of elements in the sets in question (1) above

(3).      If.

(a) A= {3,5,7,8,9,10,}, Then n(A) =

(b) B= {1, 3, 1, 2, 1, 7}, Then n(B) =

©    Q= {a, d, g, a, c, f, h, c,} , Then n(Q) =

(d) P= {4,5,6,7,…,12,13}, Then n(P) =

• D ={ days of the week} , then n(D)=
• State if the following are finite, infiniteor null set
  • Q = {x : x ≥ 7, x Є Z}

(ii)        P = {x : -4 ≤ x < 16, x Є I}

• A = {x : 2x – 7 = 2, x Є Z}
• B= { sets of goats that can fly}
• D={sets of students with four legs}

5. list the following sets in relation to the universal set. Given that the universal set ξ= {x: 1< x< 15, x ε z}

(i) A= {x; -3≤ x≤7, x ε Z}

• B= {x: 5<x<6, xεz},
• C={x: X≥ 5, xε z }

A universal set is a collection of all elements that are under consideration. In mathematics, universal sets are usually denoted by the symbol U. For example, the set of all natural numbers (N) would be considered a universal set since it contains all whole numbers including zero

Listing the elements of a universal set simply means writing out all of the elements that are contained within it. In the example above, the elements of the universal set N would be {0, 1, 2, 3, 4, 5 etc.}

1) The set of all whole numbers less than 10: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

2) The set of all prime numbers: {2, 3, 5 ,7, 11 etc.}

3) The set of all negative numbers: {-1, -2 -3 etc.}

4) The set of all fractions: {1/2, 1/3, 2/3, 1/4 etc.}

5) The set of all decimals: {0.1, 0.01, 0.001 etc.}

• list the elements of the following universal
  • The set of all positive integers
  • The set of all integers
  • ξ={ x: 1 < x < 30, x are multiples of 3}
  • ξ = { x: 7≤ x<25, x are odd numbers}
  • ξ = { x: x≥10, xεz}

More On Sets

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics, and can be used to model nearly any mathematical situation. To “solve questions on sets”, then, one must be able to understand the set-theoretic concepts involved in the question, and use these concepts to arrive at a mathematically sound answer.

Here are five examples of questions that can be solved by understanding and manipulating sets:

1. Given a set A = {1, 2, 3, 4}, find all subsets of A.

2. Find the union of the sets A = {1, 2, 3} and B = {4, 5, 6}.

3. Find the intersection of the sets A = {1, 2, 3} and B = {2, 3, 4}.

4. Given a set A = {1, 2, 3}, find the complement of A.

5. Prove that if A is a subset of B, and B is a subset of C, then A is a subset of C.

Solution

1. The subsets of {1, 2, 3, 4} are: {}, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}, and {1, 2, 3, 4}.

2. The union of {1, 2, 3} and {4, 5, 6} is {1, 2, 3, 4, 5, 6}.

3. The intersection of {1, 2, 3} and {2, 3, 4} is {2, 3}.

4. The complement of {1, 2, 3} is {4, 5, 6}.

5. A is a subset of C if and only if every element of A is also an element of C. In this case, A is a subset of B and B is a subset of C, so A is a subset of C.

Conclusion:

The subject teacher wraps up or concludes the lesson by giving out a short note to summarize the topic that he or she has just taught.

The subject teacher also goes round to make sure that the notes are well copied or well written by the pupils.

He or she makes the necessary corrections when and where the needs arise.