Understanding the Misconception: When Does a Set Contain Something?

There is often confusion surrounding the term 'contains' in set theory, as it has multiple meanings depending on the context. The English word 'contains' can refer to either an element or a subset. This article will clarify the nuances and provide clear differentiation to avoid any misunderstandings.

The Dual Meaning of 'Contains'

The problem arises because the English word 'contains' has two distinct meanings in mathematics. Sometimes it is used to denote that something is an element of a set, and other times it is used to denote that a set is a subset of another set. Because of this dual usage, it is crucial to be aware of the context to determine the correct interpretation.

Element vs. Subset

When we say, 'M contains A,' we need to clarify whether we mean (1) A is an element of M or (2) A is a subset of M.

For example, when we say, 'every set contains the empty set,' we are referring to the fact that the empty set (denoted as ?) is a subset of every set M. On the other hand, when we say, 'the set of prime numbers contains 17,' we mean that 17 is an element of the set of prime numbers P.

To avoid confusion, it is recommended to use the precise terms 'is an element of' and 'is a subset of.' For instance, instead of saying 'M contains the empty set,' we should say 'the empty set is a subset of every set M,' and instead of saying 'P contains 17,' we should say '17 is an element of the set of prime numbers P.'

Subsets and Their Characteristics

A set M contains the empty set (denoted as ?) as a subset and not as an element. This is a fundamental consequence of the definition of 'subset.'

'A is a subset of B' (denoted as A ? B) means that all elements of A are also in B. For the empty set, this is trivially true. This is because, by definition, if x ∈ ?, then x ∈ M is always true, simplifying to ? ? M.

Differences Between Subset and Element

A critical difference between A ? B and A ∈ B exists:

A ? B means that all elements of A are also in B. For example, my box of apples is a subset of the fruits, as every apple is also a fruit. A ∈ B means that A itself is an element of B. In our example, my box of apples is not a fruit itself; it is therefore not an element of the set of fruits.

The key takeaway is that A ? B indicates that A is a collection of elements all of which are found in B, while A ∈ B indicates that A is a single element of B.

Conclusion

Misunderstandings about set theory often stem from the dual meaning of the term 'contains.' By carefully considering the context and using the precise terms 'is an element of' and 'is a subset of,' we can avoid these ambiguities and ensure clarity in our mathematical communication.