Improper Subset (Definition, Symbol, Example, Venn Diagram)

Definition of Improper Subset

A set A is called Improper Subset of set B only when all the elements of set A and B are equal to each other and there is no extra element in any of the sets. Improper Subsets can simply be called Subsets. If a set is an Improper Subset of another set, then both sets are equal and have the same Cardinality.

Improper Subset Symbol

The Improper Subset Symbol is similar to the symbol of the subset. If set A is an Improper Subset of B then mathematically it can be written as,

\displaystyle A\subseteq B

Example of Improper Subset

To understand the fundamentals of Improper Subset, let us consider two sets A and B as given below.

A = {1, 2, 3}

B = {1, 2, 3}

Improper subset Venn Diagram

The above figure shows the Improper Subset Venn Diagram. From the figure, it is clear that all the elements of set A are present in the set B and both the sets are equal. There is no extra element in any of the given sets. Therefore, A is called the Improper Subset of B and vice versa. Also, A and B are called equal sets.

Difference between Improper Subset and Proper Subset | Proper subset vs Improper subset

The Differences between Improper Subset and Proper Subset are as follows:

Serial NoImproper SubsetProper Subset
1For a set A to be an Improper Subset of B all the elements of A must be present in the set B.For a set A to be a Proper Subset of B all the elements of set A must be present in set B and set B must contain at least one extra element which is not present in A.
2For a set A to be an Improper Subset of set B, A and B may be equal sets.For a set A to be a Proper Subset of B, A and B must not be equal.
3A=\{1, 2, 3, 4\}
B=\{1, 2, 3, 4\}
A is an Improper Subset of B
A=\{1, 2, 3, 4\}
B=\{1, 2, 3, 4\}
A is not a Proper Subset of B
4A=\{a, b, c, d, e\}
B=\{a, b, c, d, e, f, g\}
A is an Improper Subset of B
A=\{a, b, c, d, e\}
B=\{a, b, c, d, e, f, g\}
A is also a Proper Subset of B

Frequently Asked Questions

A set A is called Improper Subset of set B only when all the elements of set A and B are equal to each other and there is no extra element in any of the sets.

For Example,

A={a, b, c, d}

B={a, b, c, d}

All the elements of set A present in set B. Therefore, A is an Improper Subset of set B or vice versa and it can be written as,

A ⊆ B

B ⊆ A

The difference between Subset and Improper subset is:
Subset: For a set, A to be a Subset of B all the elements of set A must be present in set B and set B must contain at least one extra element which is not present in A.
Improper Subset: For a set, A to be an Improper Subset of B all the elements of A must be present in the set B.

The empty set phi is an improper subset of itself only. For any other sets, Phi is a Proper Subset.

There is no difference between the Improper subset and Equal set. For a set A to be an Improper subset of another set B, all the elements of set A must be present in set B and there must not be any extra element in the set B. Therefore, it can be concluded that the Improper Subset and Equal set are similar.

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