Difference Between Membership and Inclusion in Set Theory

Understanding the distinction between membership and inclusion is essential in set theory. Although these concepts are closely related, they have different meanings and are often confused.

• Membership
  Membership is denoted by the symbol ∈ and describes the relationship between a single element and a set. When an element x belongs to a set S, we write x ∈ S. This indicates that x is one of the elements within S.

  For instance, if S is the set {1, 2, 3}, stating 2 ∈ S is correct because 2 is an element of S. Thus, the element 2 belongs to the set S.

• Inclusion
  Inclusion is represented by the symbols ⊂ and ⊆ and refers to the relationship between two sets. Saying that a set A is included in a set B (A ⊂ B) means that all elements of A are also elements of B. Writing A ⊆ B implies that A can be exactly equal to B or a proper subset of B.

  For example, if S is the set {1, 2, 3}, stating {2} ⊂ S is correct because the set {2} is contained within S and is not equal to the entire set S. Therefore, the set {2} is included in the set S. If I wanted to be less precise and include equality, I would write {1, 2, 3} ⊆ S, which is also correct.

In summary, the membership relation tells us if an element is part of a set, while the inclusion relation tells us if one set is a subset of another.

In other words, the symbol ∈ connects an element with a set (membership), whereas the symbols ⊂ and ⊆ connect two sets (inclusion).

For example, 3 ∈ S and {3} ⊂ S are correct statements, whereas 3 ⊂ S and {3} ∈ S are incorrect.

And so on.