Proper Subset

1. What do you Mean by Proper Subset?

A proper subset is a subset of a set that contains some but not all elements of the original set. It is denoted as A ⊂ B, where A is a proper subset of B.

2. Is an Empty set a Proper Subset?

No, an empty set (∅) is not a proper subset of any set as It contains no elements .

3. Can we find Subsets of Infinite Set as Well?

Yes, you can find subsets of infinite sets. Infinite sets, like the set of natural numbers (N), can have subsets, such as the set of even natural numbers (E) or the set of prime numbers (P).

4. What is the Difference Between a Proper and an Improper Subset?

A proper subset is a subset that contains some but not all elements of the original set. An improper subset contains all elements of the original set and is the same as the original set itself.

5. Define Proper Subset.

A proper subset is a set A that contains some, but not all, elements of set B.

6. What is a Proper Subset Symbol?

The symbol for a proper subset is “⊂”. So, if set A is a proper subset of set B, it would be denoted as A⊂B.

7. What is the Difference between ⊆ and ⊂?

The symbol ⊆ represents “is a subset of or equal to,” meaning that a set A ⊆ B can be equal to set B or a proper subset. On the other hand, ⊂ represents “is a proper subset of,” indicating that a set A ⊂ B is always a proper subset, not equal to B itself.

8. What is an Example of a Proper Subset?

Consider two sets, A = {1, 2, 3} and B = {1, 2, 3, 4}. In this case, set A is a proper subset of set B. All elements in set A (1, 2, and 3) are also present in set B, but B contains an additional element (4) not found in A.

Therefore, A is a proper subset of B, denoted as A ⊂ B.

9. What are the Proper Subset of A = {1, 2, 3}?

The proper subsets of A = {1, 2, 3} are:

• {} (the empty set)
• {1}
• {2}
• {3}
• {1, 2}
• {1, 3}
• {2, 3}

Each of these subsets contains some, but not all, elements of the original set A.

10. How many Subsets does the Set A = {1, 2, 3, 4, 5} can have?

The set A = {1, 2, 3, 4, 5} have 2⁵ = 32 subsets, including the empty set and the set itself.

11. Is Null Set a Proper Subset?

No, the null set (∅) is not considered a proper subset. It is a subset of every set, but it is not a proper subset of any set because it does not contain “some” but “no” elements of the original set.

Proper Subset is a set that contains some, but not all, of the members of another set. Formally, A is considered a proper subset of B if every member of set A is also an element of set B, but there is at least one element in B that is not in A. For any proper subset, its cardinality (the number of items) is always fewer than the cardinality of the set from which it is generated.

This article covers proper subsets in detail, including their definition, examples, and symbols. In addition to that, we will also discuss improper subsets and the key differences between proper and improper subsets in this article.