Set Language and Notation: Maths Union and Intersection Explained
Set language and notation is one of those topics that students either find straightforward or find confusing, often because the symbols look unfamiliar at first glance.
The two concepts that come up most in exams are maths union and intersection. Understanding what these mean, how to write them using correct notation, and how to apply them in Venn diagram questions is essential for O-Level Math.
In these revision notes, you will learn everything you need to know about set language and notation, including:
What sets are and how to describe them
Key set notation symbols and what they mean
How maths union and intersection work
Subsets, complements and disjoint sets
Worked examples for each concept
Before you read on, you might want to download this entire revision notes in PDF format to print it out, or to read it later. This will be delivered to your email inbox.
Sets
A set is a collection of objects, things or symbols which are clearly identified. The individual objects in the set are called the elements or members of the set.
Elements may be specified in two ways:
by listing the elements
by description
Example
Listing
{2, 4, 6, 8, 10, 12}
{A, E, I, O, U}
Description
The set of even numbers between 1 and 13
The set of vowels in the alphabet
{ } These braces stand for the word “the set of”
Set Language and Notation
Set Language
‘… is an element of …’
‘… is not an element of …’
The number of elements in set A
Universal set
The empty set
Subset
A is a (proper) subset of B
A is not a (proper) subset of B
Union of sets A and B
Intersection of sets A and B
Complement of set A
Disjoint Sets
Notation
∈
∉
n(A)
U
Ø
⊆
A ⊂ B
A ⊄ B
A ∪ B
A ∩ B
A‘
A ∩ B =Ø
Naming sets and number of members in a set:
Usually, we use Capital letters to denote a set and small letters to denote members of the set.
n ( ) indicates the total number of members in a set
Example
A = {1, 3, 5, 7, 9, 11, 13} n (A) = 7
B = {2, 4, 6, 8, 10, 12} n (B) = 6
Membership of the set:
∈ is an element of (is a member of ) means (belongs to)
∉ not an element of (is not a member of) means (does not belongs to)
Finite Sets
Sets in which all the elements can be listed.
A = {1, 3, 5, 7, 9} n (A) = 5 B = {days of the week beginning with S} n (B) = 2
Infinites Sets
Sets in which it is impossible to list all the members of a set.
C = {2, 4, 6, 8, 10, . . . }
D ={x : x is a natural number}
Set Language and Notation: Relation of Sets
Universal sets: U
The set which contains all the available elements.
All proper subsets formed within the universal set draw their elements from the available elements of the universal sets.
Empty Set: { } or ø
A set which contains no elements.
An empty set in a subset of any set.
Equal Sets: ⊆
If two sets have exactly the same elements, then we say that the two sets are equal sets.
Two equal sets are also subsets (denoted by ⊆) of each other.
Example If A = {2, 4, 6, 8} and B = {8, 6, 2, 4}
then A and B are equal sets, ie. A ⊆ B or B ⊆ A
Subsets:
A ⊆ B : A is a subset of B
When every element of set A is also an element of set B, then A is a subset of B.
: A is not a subset of C
There is at least one element in the first set that does not belong to the second set.
Example If A = {2, 4, 6, 8} , B = {8, 6, 2, 4} and C = {a, b, c, d}
then A ⊆ B
Proper Subset:
A B: A is a subset of B
When each element of set A is also an element of set B, but set B has more elements than set A, then set A is a proper subset of B, denoted by “A ⊂ B”.
Therefore, in this case set B is not a proper subset of A, denoted by “ B ⊄ A”
Example Given A = {1, 5, 9} and B = {1, 3, 5, 9}
then A ⊂ B , B ⊄ A
Intersection of Sets: ∩
Common elements in different sets.
A = {1, 2, 3, 4, 5, 6}
B = {1, 3, 5, 7, 9}
A ∩ B = {1, 3, 5}
Union of Sets: ∪
The Union of set A and set B is the set of all elements which are in A, or in B, or in both A and B. It is denoted by ‘A ∪ B’ and is read as “the union of A and B”.
A = {1, 3, 4}
B = {5, 6, 7, 8}
A ∪ B = {1, 3, 4, 5, 6, 7, 8}
Complement of a set: A’
If ξ = {2, 3, 5, 7, 11, 13} and A = {2, 3, 7, 13}
A’ = {5, 11}
Disjoint Sets:
If the two sets have no element in common then the two sets are called disjoint.
The intersection of two disjoint sets is null or empty.
A = {1, 3, 5, 7}
B = {2, 4, 6, 8, 9}
A ∩ B = ø thus A and B are disjoint sets.
Frequently Asked Questions
What is the union of two sets in math?
The union of two sets, written as A ∪ B, is the set of all elements that are in A, in B, or in both. For example, if A = {1, 3, 4} and B = {5, 6, 7, 8}, then A ∪ B = {1, 3, 4, 5, 6, 7, 8}, since every element from both sets is included once.
What is the intersection of two sets in math?
The intersection of two sets, written as A ∩ B, is the set of elements that are common to both A and B. For example, if A = {1, 2, 3, 4, 5, 6} and B = {1, 3, 5, 7, 9}, then A ∩ B = {1, 3, 5}, since these are the only elements that appear in both sets.
What is the difference between a subset and a proper subset?
A is a subset of B (written A ⊆ B) when every element of A is also an element of B, and the two sets may even be identical. A is a proper subset of B (written A ⊂ B) when every element of A is in B, but B contains at least one extra element that A does not have.
What does the symbol ∈ mean in set notation?
The symbol ∈ means “is an element of” or “belongs to.” For example, if A = {1, 3, 5, 7, 9}, writing 3 ∈ A means that 3 is a member of set A. The symbol ∉ means the opposite, that an element does not belong to the set.
What are disjoint sets?
Disjoint sets are two sets that share no elements in common, so their intersection is the empty set. For example, if A = {1, 3, 5, 7} and B = {2, 4, 6, 8, 9}, then A ∩ B = ø, which means A and B are disjoint.
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