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A set is a model for an unordered collection of things in mathematics, such as integer numbers or symbols. A set can be empty, have a single element, or have infinite elements.
This lesson introduces the Notation and terminology of set theory which is basic and used throughout this course.
A set is a model of an unordered collection of things. For Example:
Students in a classroom belong to a set.
Every object in a set is called an element. For Example, a student is an element from the students’ set.
The notation x ∈ A denotes that x is an element of Set A.
The notation x ∉ A denotes that x is not an element of Set A.
In mathematics, there are three Notations to describe a set:
In the roster method notation, the elements of a set are listed inside brackets. Examples:
S = {a,d,c,b}
S = {a,b,c,d,e, f} = {a,b,c,b,c,d,e,f} , no repeated element
S = {a,b,c,d, e,f … ,z }, series
S = {{a,b,c}, r, {e,f}} , sets within a set
In the builder methods, the Set is described by representing the elements or explaining the content of the Set. For Example:
S = {x | x is a positive integer less than 50}, is a set of integers between 0 and 49
S = {x | x >= 5 or x < 0}, a set of numbers between 1 and 5
The third set notation type simply represents an interval of numbers. For Example:
[2,6] = {x | 2 ≤ x ≤ 6} , close interval, means the endpoint 2 and 6 are not included.
([2,6) = {x | 2 ≤ x ≤ 6} , openinterval, means the endpoint 2 and 6 are included.
Unless explicitly or implicitly said otherwise, all sets are assumed to be members of a fixed huge set called the universal Set, which we denote by U.
An empty Set is simply a set with no elements symbolized ∅. For Example:
s = ∅
A set contains an empty set ∅, is not empty. For Example:
S = { ∅ } is not an empty set
A Venn diagram shows the relation between a collection of sets. For example, if we have two sets, ‘A’ and ‘B’, the intersection of ‘A’ and ‘B’ is represented as the following:
Some of the commonly used sets in mathematics:
Integers set
Denoted by Z. For example: Z = {…,-5,-4,-3,-2,-1,0,1,2,3,4,5…}
Natural numbers set
Donated by N. For Example: N = {0,1,2,3,4,5….}
Positive integers set
Donated by Z⁺. For example: Z⁺ = {1,2,3,4,5…..}
Set of real numbers
Donated by R. For example: R = Set of real numbers
Set of positive real numbers
Donated by R+. For example: R+ = set of positive real numbers
Set of complex numbers
Donated by C. For Example: C = set of complex numbers.
Set of rational numbers
Donated by Q. For example: Q = Set of rational numbers
The sets below represent even integers using roster, builder, and interval notations.
Roster: {…,-4,-2,0,2,4,….}
Builder: {x | x=2n, n ∊ Z}
Interval: [a,b] = {x | a ≤ x ≤ b}
Two sets are equal if and only if they have the same elements.
A = B if \forall x (x ∊ A\leftrightarrow x ∊ B) (\forall stands for “For all”)
Examples: {1,2,3} = {3, 5, 1} , {1,2,3,3,2,1} = {1,2,3}
The set A is a subset of B, if and only if every element of A is also an element of B.
Venn diagram
If a set B is a subset of A, donated by A ⊆ B, and A is not equal to B, then A is a proper subset of B, A ⊂ B. For Example: if A = {1, 3,5} and B ={1,3}, B is a proper subset of A.
The cardinality of a set A, denoted by |A|, is the number of unique elements of A. Examples: |{1,2,3}| = 3, The set N is infinite.
Sets can be elements or subsets of other sets.
For example, if A = {4,6,{4},{8}}, then
4 is an element of A, while {4} is a set of A.
The below representation are true:
Note: ⊂ is the symbol for a proper subset, ⊄ is for ‘not a subset’
A power set of the set A, donated by P(A), is the set of all subsets of A
Example: If A = {a,b} then
P(A) = {ø, {a},{b},{a,b}}
The total number of subsets = | P(A) | = 2|A| (power a)
Example, the number of subsets P(A) if A ={a,b) is 2 power 2 = 4.
The Cartesian product of two sets A and B (A x B), is the set of all possible ordered pairs (a,b), where a ∈ A and b ∈ B:
For example:
If A = {1,2,3} and B = {a,b}
The cartesian product:
A x B = {(1,a),(2,a),(3,a),(1,b),(2,b),(3,b)}
B x A = {(a,1), (b,1),(a,2).(b,2),(a,3),(b,3)}
Consider the following sets:
A = {a,b,c} , B={1,2}, and C={5}, then:
sets