Improper Subset

1.0Introduction to Improper Subset

The concept of subsets is fundamental in Set Theory, an important topic for JEE Mathematics. Subsets form the foundation of many advanced concepts, such as relations, functions, probability, and algebraic structures.

While studying subsets, we encounter two types: Proper Subsets and Improper Subsets. Many JEE aspirants often overlook improper subsets, considering them trivial, but they are crucial for theoretical clarity and multiple-choice problems in JEE.

This guide explains improper subsets in detail, including definitions, examples, properties, Venn diagram representation, and their application in JEE-level problems.

2.0Definition of Improper Subset

A set A is said to be an improper subset of itself if all the elements of A are contained in A, and there is no extra element left out.

 In simple words: A subset is called an improper subset if it is equal to the original set itself.

Example:

• Let A = {1, 2, 3}
• Subsets of A = {∅, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}}
• Among these, {1,2,3} = A, which is the improper subset of A.

Thus, every set has exactly one improper subset, which is the set itself.

3.0Notation of Improper Subset

The symbol for subset is ⊆.

• A ⊆ A → A is an improper subset of A.
• If A ⊂ B → A is a proper subset of B.

Remember: ⊆ allows equality, while ⊂ does not.

4.0Key Properties of Improper Subsets

1. Every set is an improper subset of itself: A ⊆ A always holds true.
2. Only one improper subset exists for any set.: Example: If A = {1, 2, 3}, then A itself is the only improper subset.
3. The empty set (∅) is never an improper subset : ∅ is always a proper subset, not an improper one.
4. Relation with Power Set: In the power set P(A) of a set A, the set A itself is the improper subset.
5. Connection with Superset: If A ⊆ B and A = B, then A is an improper subset of B, and B is an improper superset of A. 

5.0Examples of Improper Subsets 

Example 1:

Let A = {a, b}.

• Subsets = {∅, {a}, {b}, {a,b}}
• Improper subset = {a,b}.

Example 2:

Let X = {1, 2, 3, 4}.

• Total subsets = 2⁴ = 16.
• Among these, the improper subset is X itself = {1, 2, 3, 4}.

Example 3:

Let B = {x}.

• Subsets = {∅, {x}}
• Improper subset = {x}.

6.0Improper Subset Problems

Problem 1: Find the number of improper subsets of A = {1,2,3,4,5}.

Solution:

• By definition, a set has only 1 improper subset (the set itself).
• Answer: 1

Problem 2: If X has 10 elements, how many improper subsets does X have?

Solution:

• No matter how large the set is, the number of improper subsets = 1.
• Answer: 1

Problem 3: If A has 3 elements, find the number of proper and improper subsets.

Solution:

• Total subsets = 2³ = 8
• Proper subsets = 8 – 1 = 7
• Improper subsets = 1
• Answer: 7 proper, 1 improper.

Problem 4: Let U = {1,2,3,4,5}, A = {2,3,4,5}. Is A an improper subset of U?

Solution:

• Since A ≠ U, A is only a proper subset of U.
• A is not an improper subset.
• Answer: No.