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

Proper Subset

What is Proper Subset?

A set A is called a Proper Subset of B only when

  • Set A and set B are not equal and
  • Set A is a subset of set B.

Proper Subset Symbol

If set A is a Proper Subset of set B, then mathematically, it can be written as,

A\subset B

Proper Subset Example

Let us consider two sets A and B as given below.

A = {1, 2, 3}

B = {1, 2, 3, 4, 5}

Proper subset venn diagram

The figure shows the Venn diagram of a Proper Subset. Here, Set A is called the Proper Subset of set B because all the elements of set A are present in the set B, and set A and B are not equal.

Therefore,

\displaystyle A\subset B.

Frequently Asked Questions (FAQ)

A set A is called proper subset of set B when all the elements of A are present in set B and set A and B are not equal. However, set A is called improper subset of set B when set A and B are equal sets.

Yes, a set can be a subset and a proper subset at the same time. To be a proper subset, both sets should not be equal.

For a set to be a proper subset of any set A, the set must have at least one element of set A. But, a null set does not have any element. Therefore, a null set is not a proper subset.

The set phi is called an empty set. An empty set is a proper subset of any set. Therefore, Phi is a proper subset.

There is no difference between Improper subset and Equal set. This is because, for a set to be an Improper subset of any set all the elements of both the sets should be equal.

There is only one improper subset of a set because, for a set to be a improper subset of any set, both the sets must be equal.

The set phi is called an empty set. An empty set is a proper subset of any set. Phi is not a improper subset.

The sign of improper subset is the same as a subset and is denoted by

The Improper subset of of set A={a, b, c, d} is the set itself i.e., {a, b, c, d}.

An empty set Phi is always a proper subset. Therefore, empty set Phi is not a improper subset of any set. However, an empty set can be an improper subset of itself.

A set A is called Improper Subset of  only when all the elements of set  and  are equal. Improper Subsets can simply be called Subsets. If a set is an Improper Subset of another set, then both sets are equal.

A set A is called a Proper Subset of B only when
Set A and set B are not equal and
Set A is a subset of set B.

  • Let A={1, 2, 3, 4} and B={1, 2}. Find out if B is a proper subset of A.
  • Check whether all the elements of B are present in A. The elements of B are 1 and 2. Both are present in A.
  • Check whether A and B are equal. The sets A and B are not equal as elements 3 and 4 are not present in the set B.
  • If both the conditions in step 2 and 3 are satisfied then B is a proper subset of A. Both the conditions are satisfied. Therefore, B is a proper subset of A.

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