Proper Subset vs. Subset

Attribute	Proper Subset	Subset
Definition	A set A is a proper subset of a set B if every element of A is also an element of B, and A is not equal to B.	A set A is a subset of a set B if every element of A is also an element of B, and A may be equal to B.
Symbol	A ⊂ B	A ⊆ B
Example	{1, 2} is a proper subset of {1, 2, 3}	{1, 2} is a subset of {1, 2, 3}

Definition

A subset is a set that contains all the elements of another set. In other words, if every element of set A is also an element of set B, then A is a subset of B. This relationship is denoted as A ⊆ B. On the other hand, a proper subset is a subset that is not equal to the original set. In other words, if every element of set A is also an element of set B, but there exists at least one element in set B that is not in set A, then A is a proper subset of B. This relationship is denoted as A ⊂ B.

Example

For example, let's consider two sets: A = {1, 2, 3} and B = {1, 2, 3, 4}. In this case, set A is a subset of set B because all the elements of A are also present in B. However, A is not a proper subset of B because they are equal. If we consider another example where A = {1, 2} and B = {1, 2, 3}, then A is a proper subset of B because all the elements of A are present in B, but B has an additional element, 3, which is not in A.

Cardinality

One key difference between a subset and a proper subset is the cardinality of the sets. The cardinality of a set is the number of elements in the set. When A is a subset of B, it is possible for both sets to have the same number of elements, making them equal in cardinality. However, when A is a proper subset of B, the cardinality of A will always be less than the cardinality of B since B has at least one element that A does not have.

Relationship

Another important aspect to consider when comparing proper subsets and subsets is the relationship between the sets. In the case of a subset, the relationship is inclusive, meaning that the subset can be equal to the original set. This implies that every element of the subset is also an element of the original set. On the other hand, in the case of a proper subset, the relationship is exclusive, meaning that the proper subset cannot be equal to the original set. This implies that while every element of the proper subset is in the original set, there is at least one element in the original set that is not in the proper subset.

Notation

Proper subsets and subsets are denoted using specific symbols in set theory. The symbol ⊆ is used to represent a subset, indicating that every element of the first set is also an element of the second set. The symbol ⊂ is used to represent a proper subset, indicating that every element of the first set is also an element of the second set, but the two sets are not equal. These symbols help to clarify the relationship between sets and distinguish between subsets and proper subsets.

Usage

Understanding the difference between proper subsets and subsets is important in various mathematical contexts. In set theory, these concepts are fundamental to defining relationships between sets and understanding their properties. Proper subsets are often used to describe strict containment, where one set is completely contained within another set without being equal to it. Subsets, on the other hand, are used to describe a broader relationship where one set includes all the elements of another set, possibly including additional elements as well.

Conclusion

In conclusion, proper subsets and subsets are related concepts in set theory that describe the relationship between two sets. While a subset includes all the elements of another set, a proper subset is a subset that is not equal to the original set. The cardinality, relationship, notation, and usage of proper subsets and subsets all play a role in distinguishing between the two concepts. Understanding these distinctions is essential for working with sets and analyzing their properties in mathematics.