TOPIC: SETS
Content:
Definition of sets
Set notations
Types of sets
Definition
A set is a well-defined list or collection of objects with some characteristics which are unique to its members. Examples:
(i) a set of mathematics text books
(ii) a set of cutleries
(iii) a set of drawing materials etc.
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 member or element of the set.
SET NOTATION
This is a way of representing a set using any of the following.
(i) Listing method
(ii) Rule method or word description
(iii) Set builders notation.
https://youtu.be/NX67x1d4k5Y
(i) 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.
(ii) Rule Method.
The elements in a set can be defined also by describing the rule or property that connects its members. 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
(iii) 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 numbers.
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 11.
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.
https://youtu.be/tyDKR4FG3Yw
EVALUATION
(a) Define Set
(a) C = {x : 3x – 4 = 1, x є ƶ}
(b) P = {x : x is the prime factor of the LCM of 15 and 24}
© Q = {The set of alphabets}
(d) R = {x : x ≥ 5, x is an odd number}
Set – Builders Notations (contd.)
Examples 1:
List the elements of the following sets
(i) A = {x : 2 < x ≤ 7, x є ƶ}.
(ii) B = {x : x > 4, x є ƶ}
(iii) 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}
(vii) P = {x : 15 ≤ x < 25, x are numbers divisible by 3}
(viii) 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).
(iv) To be able to list the elements of this set, 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}
(v) We also need to solve the equation to get the set values
3x - 2 = x + 3
3x – x = 3 + 2
2x = 5
∴ x = 5/2
Since 5/2 is not an integer (whole number) therefore the set will contain no element.
∴є = { } or Ø
(vi) 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 falls within 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:
(i) 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}
(ii) B = {x : x > 2, x є ƶ} OR
B = {x : x ≥ 3, x є ƶ}
(iii) C = {x : x < 25, x є ƶ} OR
C = {x : x ≤ 24, x є ƶ}
(iv) D = {x : x > 8 or x ≥ 7, x is odd, x є ƶ}
(v) P={1,2} suggests the solutions of a quadratic equation. 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}
(vi) Q = {x : x is a vowel}
https://youtu.be/xnfUZ-NTsCE
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 notations.
(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}
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:
(i) B = {x : x ≤ 7, x є ƶ}
(ii) 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.
https://youtu.be/VBzlvKP-2yI
https://youtu.be/WL46UtTPr_c
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) =
(e) D ={ days of the week} , then n(D)=
(4) State if the following are finite, infinite or null set
(i) Q = {x : x ≥ 7, x Є Z}
(ii) P = {x : -4 ≤ x < 16, x Є I}
(iii) A = {x : 2x – 7 = 2, x Є Z}
(iv) B= { sets of goats that can fly}
(v) 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}
(ii) B= {x: 5<x<6, xεz},
(iii) C={x: X≥ 5, xε z }
(6) list the elements of the following universal sets.
(i) The set of all positive integers
(ii) The set of all integers
(iii) ξ={ x: 1 < x < 30, x are multiples of 3}
(iv) ξ = { x: 7≤ x<25, x are odd numbers}
(v) ξ = { x: x≥10, xεz}
ASSIGNMENT
New General Mathematics for SSS, Book 1 Pages 97- 100 Exercise 8c
Question no. 1,4 and 5
READING ASSIGNMENT
1. New General Mathematics for SSS, Book 1 Pages 97 – 100.
2. Man Mathematics for SSS, Book 1, Pages 45 - 60
