Types of Subset | Difference Between a Proper And Improper Subset

If all members of set B are present in set C then we say that set B is a subset of set C. We know that a set is a well-defined collection of different numbers or any items. If set B = {5,6,7} and set C = {4,5,6,7,8,9} we can say that set B is a subset of set C since all the members in set B are present in set C.in this article learn about the types of a subset.

Types of subset

There are two types of a subset

• Proper subset
• Improper subset

Proper subset

A proper subset is any subset of the set except itself a set. We know that every set is a subset of itself but it is not also said of a proper subset of itself.

For example:

If A = {a, b, c}, then its proper subsets are {}, {a}, {b}, {c}, {a, b}, {a, c}, and {b, c}, but the set itself {a, b, c} is not a proper subset of A.

Symbol proper subset

The proper subset symbol is ⊂. i.e., if A is a proper subset of B, then: A ⊂ B and A ≠ B

Formula proper subset

The number of proper subsets of a set with ‘n’ members is 2ⁿ – 1. The number of subsets of a set with ‘n’ members is 2ⁿ. We also know all the subsets of a set except the set itself are the proper subsets of the set, the number of proper subsets is obtained by subtracting 1 from 2ⁿ.

For example:

• The number of proper subsets of A = {a, b} is,

 2² – 1 = 4-1=3

• The number of proper subsets of set A = {1, 2, 3,4} is,

 2⁴ – 1 = 16-1=15

• The number of proper subsets of empty set Φ = { } is, 2⁰ – 1 = 1-1=0.

Improper subset

A set A is called an Improper Subset of set B only when all the members of set A and B are equal to each other and there is no extra element in any of the sets. Improper Subsets can be also said to be Subsets. If a set is an Improper Subset of another set, then both sets are equal and have the same Cardinality. Every set A has only one improper subset which is set A itself.

 Examples of improper subsets.

• {a, b, c} is the only improper subset of itself a set

{a, b, c}

• {3, 4} is the only improper subset of itself set {3, 4}

Symbol of an improper subset

Improper subset symbol ⊆ If set A is an Improper Subset of B then mathematically it can be written as A ⊆ B and A=B “⊆” means “subset or equal to”

Improper subset formula

The number of improper subsets of a set with ‘n’ members is always 1. i.e., the number of improper subsets of a set does not depend upon the number of members of the set.

 Example of an improper subset

• {a, b, c,d} is the only improper subset of {a, b, c,d}
• {1, 2, 3} is the only improper subset of itself a set

 {1, 2, 3}

Difference between a proper and improper subset

Proper subset	improper subset
Proper subset It contains only a few elements of set A.	Improper subset It contains all elements of set A.
The number of a proper subset of A used formula is 2n – 1.	The number of improper subsets of A is just 1 (which is A itself).
The symbol “⊂” should be used only for proper subsets.	The symbol for improper subsets. Can be used in⊆
Proper subset It is never equal to set A.	The improper subset is always equal to set A.