# 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**

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 2^{n} – 1. The number of subsets of a set with ‘n’ members is 2^{n}. 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 2^{n}.

**For example:**

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

2^{2} – 1 = 4-1=3

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

2^{4} – 1 = 16-1=15

- The number of proper subsets of empty set Φ = { } is, 2
^{0}– 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 2^{n }– 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. |

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