Introduction

1. The following shows the summary of set language and notation.
∈ - is an element of 
∉ - is not an element of
⊆ - is a subset of 
⊊ - is not a subset of
⊂ - is a proper subset of 
⊄ - is not a proper subset of 
n(A) - number of elements in set A
ξ - universal set
Ø or {} - empty set, or null set
A’ - complement of set A
A ∪ B - union of A and B
A ∩ B - intersection of A and B 

2. Each object in a set is called the element/member of the set. 

3. Three ways to describe a set are shown in the table below. 

Further Explanation/Examples (Elements)

1. If x is an element of set A, we write x ∈ A.
If x is not an element of set A, we write x ∉ A.

E.g. A = {1, 3, 5, 7}
1 ∈ A, 3 ∈ A but 2 ∉ A. 

2. An empty/null set is a set containing no elements. It is denoted by Ø or {}.

E.g. A = {x : x is a natural number < 1}
Hence A = Ø, since natural numbers are 1, 2, 3, 4, ...

3. The number of elements of set A is denoted by n(A).

E.g. 1: A = {2, 4, 6, 8}
       n(A) = 5
E.g. 2: B = {x : x is an odd number} = {1, 3, 5, 7, 9, ...}
        n(B) = infinite

Further Explanation/Examples (Sets)

1. Two sets are equal if they contain exactly the same elements. If two sets A and B are equal, we write it as A = B. If two sets A and B are not equal, we write it as A ≠ B.

E.g. 1: C = {x : x is a letter of the word ’apple’} = {a, p, l, e}
           D = {x : x is a letter of the word ’leap’} = {l, e, a, p}
Hence C = D.

E.g. 2: E = {x : x + 1 = 3} = {2}
          F = {x : x2 = 4} = {2, -2}
Hence E ≠ F.

2. If every element of set A is also an element of set B, then set A is a subset of set B and is written as A ⊆ B. 
If set A is not a subset of set B, we write as A ⊊ B.

E.g. A = {2, 4, 5}
       B = {1, 2, 3, 4, 5}
A ⊆ B since every element of A is also an element of B.
B ⊊ A since not every element of B is an element of A. 
*Note: If A ⊆ B and B ⊆ A, then A and B have exactly the same elements, i.e. A = B.

3. Set A is a proper subset of set B if every element of set A is also an element of set B and set B has more elements than set A. We write A ⊂ B to denote A is a proper subset of B. 
E.g. A = {1, 2, 3}
       B = {1, 2, 3, 4}
       A ⊂ B

4. The Universal set, denoted by ξ is the set that contains all elements being considered in a given discussion.
E.g. A = {2, 4, 6, 8}
       B = {1, 3, 5, 7, 9}
The Universal set for A and B could be:
ξ = {x : x is a natural number less than 10}

Note: The empty set, Ø, is a subset of every set. For any set A, Ø ⊆ A ⊆ ξ.

The following is a recap of what some expressions mean:

Introduction of Venn Diagrams

1. In a Venn diagram, 
- a large rectangle is used to represent the Universal set ξ.
- circles/ovals are drawn inside the rectangle to represent the subsets of ξ. 

2. The complement of set A, written as A’, is the set of all elements in the Universal set ξ that are not in A. We read A’ as ’A complement’ or ’A prime’. 

The following image shows an example of a Venn diagram: 

Video Explanation

Watch the following video for an introduction to what Venn diagrams are and what they can represent:

Union and Intersection of Sets

1. The union of two sets, A and B, is the set of elements which are in A or in B or in both A and B. It is denoted by A ∪ B. 
2. The intersection of two sets, A and B, is the set of elements which are common to both A and B. It is denoted by A ∩ B. 
3. If two sets A and B are disjoint sets, then A ∩ B = Ø. 

Video Explanation

The following video further explains the concept of Union and Intersection when it comes to Venn diagrams:

Drawing Venn Diagrams

The following video gives a more in-depth explanation and more examples of drawing Venn diagrams: