A Set is a well defined collection of objects. The term 'Well defined' means that it is possible to tell whether a given object belongs to the set or not. The students in your class from one set, the months of the year from another.
In mathematics, sets are usually denoted by capital letters as A, S or X. The objects that make up a set are referred to as its elements or members. elements of a set are normally denoted by lower-case letters such as a, s or x.
We can indicate the members of a set in various ways. If the set contains only a small number of elements, we may simply list them in any order within curly brackets. for example:
A= {2,4,6}
When a set contains a large or perhaps infinite number of elements and the members of the set exhibit an obvious pattern, we may indicate this pattern with three dots, which mean 'and so on' or 'and so on up to'. For example:
B= {3,6,9,12,...}
is the set of all positive multiples of 3, and contains an infinite number of elements, while:
C= {a,b,c,...,x,y,z}
is the set of letters in the alphabet.
You should familiarise yourself with the following standard notation for sets.
x ∈ A
This is read as ' x is a member of (or belongs to) A '.
Example
If V is the set of vowels in the alphabet, then the letter ' u ' is a member of V.
u ∈ V
Example
If A is the set { 2, 3, 5, 7, 11 } and x = 5, then :
x ∈ A
x ∉ A
This is read as ' x is not a member of (or does not belong to) A '.
Example
If V is the set of vowels in the alphabet, then the letter ' g ' is not a member of V.
g ∉ V
Example
If A is the set { 2, 3, 5, 7, 11 } and x = 5 . 4, then :
x ∉ A
A ⊂ B
This is read as ' A is a subset of B ' and means that every member of the set A is a member of the set B.
Example
If M is the set of months in the year and T is the set of months having 30 days, then T is a subset of M.
T ⊂ M
Example
If A is the set { 3, 4, 5 } and B is the set {1, 2, 3, 4, 5, 6, 7} then:
A ⊂ B
A ∪ B
This is read as ' A union B ' and is the set of all elements that are in A or in B or both.
Example
If T is the set of students who travel by train, B is the set of students who travel by bus and P is the set of students who travel by train or bus, then P is the union of sets T and B.
P = T ∪ B
Example
If A is the set {a, b, c, d} and B is the set {b, d, e, f } then:
A ∪ B = {a, b, c, d, e, f }
A ∩ B
This is read as ' A intersect B ' and is the set of all elements that are in both A and B.
Example
If R is the set of students who play rugby, T is the set of students who play tennis and B is the set of students who play rugby and tennis, then B is the intersection of sets R and T.
B = R ∩ T
Example
If A is the set {a, b, c, d} and B is the set {b, d, e, f } then:
A ∩ B = {b, d}
The main purpose of this brief introduction to set theory is to enable us to refer to various sets of numbers using the following standard notations and terminology.
N
The set of all natural numbers, also referred to as the positive integers.
N = {1, 2, 3, 4, ... }
Z
This is the set of all integers
Z = {..., -3, -2, -1, 0, 1, 2, 3, ... }
Q
The set of rational numbers.
A rational number is a number that can be expressed as a ratio of two integers. e.g., 1/2, -7/4, 9/2. Note that
integers are rational. e.g., 3 = 3/1.
Q = {p/q | p ∈ Z and q ∈ N }
where the bar | is read as 'with the property' or 'such that'. There are numbers that are not rational. These numbers are called irrational numbers.
We cannot express these numbers as a ratio of two integers.
For example, √2 and π are irrational numbers.
R
The set of real numbers.
Unless you have studied complex numbers, then all numbers that you will have come across in mathematics so far are real numbers. Every point on the number line represents a real number.
[a, b]
This is the closed interval:
{x ∈ R | a ≤ x ≤ b}
Here a and b are real numbers and are called the endpoints. They are included in the closed interval, and this is often indicated by a filled-in dot at the ends of the interval on the number line.
(a, b)
This is the open interval:
{x ∈ R | a < x < b}
Again, a and b are real numbers and are called the endpionts. They are not included in the open interval and this is often indicated by an open dot at the ends of the interval on the number line.
Sometimes we can have:
∞ or -∞
Note that the infinity symbols do not represent real numbers, they are used to indicate that the set is unbounded in the positive or negative direction of the real number line.
(-∞, b) = {x ∈ R | x < b}
(-∞, b] = {x ∈ R | x ≤ b}
(a, ∞) = {x ∈ R | x > a}
[a, ∞) = {x ∈ R | x ≥ a}
∃
'There exists'
e.g.,
∃ x ∈ R
such that x2 = 2
∀
'For all'
e.g.,
∀ x ∈ R, x2 > 0
In mathematics, sets are usually denoted by capital letters as A, S or X. The objects that make up a set are referred to as its elements or members. elements of a set are normally denoted by lower-case letters such as a, s or x.
We can indicate the members of a set in various ways. If the set contains only a small number of elements, we may simply list them in any order within curly brackets. for example:
A= {2,4,6}
When a set contains a large or perhaps infinite number of elements and the members of the set exhibit an obvious pattern, we may indicate this pattern with three dots, which mean 'and so on' or 'and so on up to'. For example:
B= {3,6,9,12,...}
is the set of all positive multiples of 3, and contains an infinite number of elements, while:
C= {a,b,c,...,x,y,z}
is the set of letters in the alphabet.
You should familiarise yourself with the following standard notation for sets.
x ∈ A
This is read as ' x is a member of (or belongs to) A '.
Example
If V is the set of vowels in the alphabet, then the letter ' u ' is a member of V.
u ∈ V
Example
If A is the set { 2, 3, 5, 7, 11 } and x = 5, then :
x ∈ A
x ∉ A
This is read as ' x is not a member of (or does not belong to) A '.
Example
If V is the set of vowels in the alphabet, then the letter ' g ' is not a member of V.
g ∉ V
Example
If A is the set { 2, 3, 5, 7, 11 } and x = 5 . 4, then :
x ∉ A
A ⊂ B
This is read as ' A is a subset of B ' and means that every member of the set A is a member of the set B.
Example
If M is the set of months in the year and T is the set of months having 30 days, then T is a subset of M.
T ⊂ M
Example
If A is the set { 3, 4, 5 } and B is the set {1, 2, 3, 4, 5, 6, 7} then:
A ⊂ B
A ∪ B
This is read as ' A union B ' and is the set of all elements that are in A or in B or both.
Example
If T is the set of students who travel by train, B is the set of students who travel by bus and P is the set of students who travel by train or bus, then P is the union of sets T and B.
P = T ∪ B
Example
If A is the set {a, b, c, d} and B is the set {b, d, e, f } then:
A ∪ B = {a, b, c, d, e, f }
A ∩ B
This is read as ' A intersect B ' and is the set of all elements that are in both A and B.
Example
If R is the set of students who play rugby, T is the set of students who play tennis and B is the set of students who play rugby and tennis, then B is the intersection of sets R and T.
B = R ∩ T
Example
If A is the set {a, b, c, d} and B is the set {b, d, e, f } then:
A ∩ B = {b, d}
The main purpose of this brief introduction to set theory is to enable us to refer to various sets of numbers using the following standard notations and terminology.
N
The set of all natural numbers, also referred to as the positive integers.
N = {1, 2, 3, 4, ... }
Z
This is the set of all integers
Z = {..., -3, -2, -1, 0, 1, 2, 3, ... }
Q
The set of rational numbers.
A rational number is a number that can be expressed as a ratio of two integers. e.g., 1/2, -7/4, 9/2. Note that
integers are rational. e.g., 3 = 3/1.
Q = {p/q | p ∈ Z and q ∈ N }
where the bar | is read as 'with the property' or 'such that'. There are numbers that are not rational. These numbers are called irrational numbers.
We cannot express these numbers as a ratio of two integers.
For example, √2 and π are irrational numbers.
R
The set of real numbers.
Unless you have studied complex numbers, then all numbers that you will have come across in mathematics so far are real numbers. Every point on the number line represents a real number.
[a, b]
This is the closed interval:
{x ∈ R | a ≤ x ≤ b}
Here a and b are real numbers and are called the endpoints. They are included in the closed interval, and this is often indicated by a filled-in dot at the ends of the interval on the number line.
(a, b)
This is the open interval:
{x ∈ R | a < x < b}
Again, a and b are real numbers and are called the endpionts. They are not included in the open interval and this is often indicated by an open dot at the ends of the interval on the number line.
Sometimes we can have:
∞ or -∞
Note that the infinity symbols do not represent real numbers, they are used to indicate that the set is unbounded in the positive or negative direction of the real number line.
(-∞, b) = {x ∈ R | x < b}
(-∞, b] = {x ∈ R | x ≤ b}
(a, ∞) = {x ∈ R | x > a}
[a, ∞) = {x ∈ R | x ≥ a}
∃
'There exists'
e.g.,
∃ x ∈ R
such that x2 = 2
∀
'For all'
e.g.,
∀ x ∈ R, x2 > 0
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