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# Types of Subset | Difference Between a Proper And Improper Subset

August 27, 2022 written by Azhar Ejaz

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

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}

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