Understanding Set Subsets: When A Equals B

In set theory, the concept of subsets is fundamental to many mathematical proofs and logical arguments. One common question is: If set A equals set B, is A a subset of B? This article aims to clarify this idea and provide a deeper understanding of sets and their relationships.

What is a Subset?

A set A is considered a subset of a set B if every element of A is also an element of B. This relationship is denoted as ( A subset B ).

Definition 1: Traditional Subset

Formally, a set A is a subset of a set B if and only if there is no element of A that is not an element of B. This can be mathematically expressed as:

( A subset B ) if and only if for all ( x ), ( x in A implies x in B ).

Definition 2: Alternative Subset

Another way to express this is that every element of A is also an element of B. This definition is equivalent to the first but can sometimes be misinterpreted to imply that A must contain at least one element and that all elements of A must be elements of B. This concern is addressed by the first definition, which explicitly covers the case of the empty set.

What if A Equals B?

If set A equals set B, denoted as ( A B ), then A is indeed a subset of B, and B is a subset of A. This can be intuitively understood as follows:

If A B:

A is a subset of B because every element of A is an element of B (since A and B are exactly the same).

B is a subset of A because every element of B is an element of A (for the same reason).

Thus, if A B, then ( A subset B ) and ( B subset A ).

Proper Subset

A set A is a proper subset of a set B if A is a subset of B but B is not a subset of A. This can be denoted as ( A subsetneq B ). The key difference is that there must be at least one element in B that is not in A.

Equivalence

Set A and set B are said to be equivalent if they have exactly the same elements. Mathematically, this is the same as saying that ( A B ). One way to define the equivalence of sets is through the subset relationship:

Theorem: A B if and only if ( A subset B ) and ( B subset A ).

This theorem shows that if both ( A subset B ) and ( B subset A ), then A and B must have the same elements, thus A B.

Confusion: Intuitive Size Differences

A common confusion is the intuitive idea that if ( A subset B ), then A must be a smaller set. However, in set theory, the subset relation does not imply a size difference. It only implies a structural relationship between the elements of the sets. For example, the set of natural numbers is a subset of the set of real numbers, but they are not the same in size. This distinction is crucial in advanced mathematical discussions.

In summary, when A equals B, A is a subset of B, and B is a subset of A. This is due to the definition of subset and the transitive property of equality. Understanding these relationships is key to advancing in set theory and related mathematical fields.