Empty Set or Null Set: A set which does not contain any element is called an empty set, or the null set or the void set and it is denoted by and is read as phi.
In roster form, ∅ is denoted by { }.
An empty set is a finite set,
For example: (a) The set of whole numbers less than 0.
Note:∅ ≠ {0}
The cardinal number of an empty set, i.e., n(∅) = 0
∴ ∅ has no element.
{0} is a set which has one element 0.
Singleton Set: A set which contains only one element is called a singleton set.
Ex A = {x : x is neither prime nor composite}
B = {x : x is a whole number, x < 1} is a singleton set containing one element, i.e., 1.
This set contains only one element 0 and is a singleton set.
• Let A = {x : x ∈ N and x² = 4}
Here A is a singleton set because there is only one element 2 whose square is 4.
• Let B = {x : x is a even prime number}
Here B is a singleton set because there is only one prime number which is even, i.e., 2.
UNIVERSAL SET:
A universal set is a set which contains all objects including itself.
All sets under consideration will be subsets of a fixed set. We call this set the Universal Set.
A set which contains all the elements of other given sets is called a universal set.
The symbol for denoting a universal set is or ξ.
Subset:- If A and B are two sets, and every element of set A is also an element of set B, then A is called a subset of B, and we write it as A ⊂ B or B ⊂ A
If all elements of set A are present in B , then A is said to be subset of B denoted by A ⊂ B
The symbol ⊂ stands for ‘is a subset of’ or ‘is contained in’
• Every set is a subset of itself,
i.e., A ⊂ A, B ⊂ B.
• Empty set is a subset of every set.
• Symbol ‘⊆’ is used to denote ‘is a subset of’ or ‘is contained in’.
• A ⊆ B means A is a subset of B or A is contained in B.
• B ⊆ A means B contains A.
For example;
1. Let A = {2, 4, 6} B = {6, 4, 8, 2} Here A is a subset
of B
2. Notes: If A⊂ B and B⊂ A, then A = B, i.e., they are equal sets.
Super Set: Let A and B are two sets. If A⊂ B , then B is called the super set of A.
Whenever a set A is a subset of set B, we say that B is a superset of A and we write, B ⊇ A.
Symbol ⊇ is used to denote ‘ is a super set of ’
For example;
A = {a, e, i, o, u} B = {a, b, c, …………., z}
Here A ⊆ B i.e., A is a subset of B but B ⊇ A i.e., B is a super set of A