Question

Example 6 shows that \(\varnothing\) is the complement of \(U\). But since the null set is a subset of any set, \(\varnothing\) must be a subset of \(U\). Inasmuch as the term "complement of \(U^{\prime \prime}\) implies the notion of being not in \(U\), whereas the term "subset of \(U^{\prime \prime}\) implies the notion of being in \(U,\) it seems paradoxical for \(\varnothing\) to be both of these. How do you resolve this paradox?

Short answer

The paradox is resolved by recognizing different concepts: \( \varnothing \) as a complement of \( U \) means no elements are outside \( U \), and as a subset, it does not contradict this since it contains no elements.

Step-by-step solution

Understanding Complements and Universal Sets

The universal set, denoted as \( U \), contains all possible elements within a particular context. The complement of a set \( A \), written as \( A' \), consists of all elements not in \( A \) but in \( U \). For \( U \), the complement is the null set \( \varnothing \) because there are no elements outside \( U \).

Reviewing Null Set as Subset

The null set \( \varnothing \) is defined to be a subset of any set, including the universal set \( U \). This is because a subset \( A \) of \( B \) means every element of \( A \) is in \( B \). Since \( \varnothing \) has no elements, it trivially satisfies this condition for any set, including \( U \).

Clarifying the Apparent Paradox

The apparent paradox arises from confusing two different types of relationships: being a subset (\( \varnothing \subseteq U \)) and being a complement (\( U' = \varnothing \)). The term 'complement' indicates elements not present, while 'subset' refers to a particular relationship of having all elements within another set. \( \varnothing \) being a subset of \( U \) is about inclusion, and \( \varnothing \) being the complement of \( U \) is about what is outside \( U \).

Resolving the Paradox

The resolution is that \( \varnothing \) as the complement of \( U \) means that outside \( U \), there are no elements (because \( U \) is the universal set). As a subset, \( \varnothing \subseteq U \) holds true since it contains no elements that violate the subset condition. The terms describe different aspects and do not conflict when accurately understood.

Key concepts

Universal Set

In the realm of set theory, a Universal Set, often denoted as \( U \), is a comprehensive collection that includes every possible element for a specific discussion. Think of it as the complete universe of discourse or context that encompasses all elements under consideration. For instance, if you're discussing all kinds of fruit in a grocery store, then your Universal Set would include apples, bananas, oranges, etc. When we refer to the Universal Set, it forms the backdrop against which all other sets are defined. This makes it unique as it does not have any elements outside of itself. In every conversation or mathematical problem using sets, the Universal Set helps in defining the boundaries within which other sets operate.

• It is crucial for defining complements, as complements are simply elements not found in the specific subset but within this universal boundary.
• Because of its containing nature, the concept of a complement for \( U \) is represented by the Null Set, since nothing exists outside of \( U \).

Null Set

The Null Set, symbolized as \( \varnothing \), is a fundamental concept in set theory. It represents a set with no elements. Just like an empty box, it contains nothing but is still recognized as a set. The fascinating aspect of the Null Set is its property of being a subset of every possible set, including the Universal Set \( U \). Why is this? Because the definition of a subset states that all elements of one set should also be elements of another. Since the Null Set has no elements, it automatically meets this condition for any set.

• It is crucial in different mathematical operations and helps simplify the understanding of vacuous truths.
• The Null Set is unique, as it is the only set with this innate attribute.

Subset

A Subset describes a scenario where every element of one set is also contained within another set. When we say set \( A \) is a subset of set \( B \), it's denoted as \( A \subseteq B \), indicating all elements of \( A \) are in \( B \) too. This relationship is foundational in understanding how different sets can be related. For example, if \( A = \{1, 2\} \) and \( B = \{1, 2, 3\} \), we see that \( A \subseteq B \).

• Subsets allow researchers and mathematicians to formulate concepts of inclusion and hierarchy within sets.
• Any set is a subset of itself, and the Null Set is a subset of every possible set, maintaining consistency in logical applications.

Complement

The Complement of a set \( A \), written as \( A' \), refers to all the elements not contained within \( A \) but present in the Universal Set \( U \). This offers a way to look at what is outside a set in the context of \( U \). If \( U = \{1, 2, 3, 4\} \) and \( A = \{1, 2\} \), then \( A' = \{3, 4\} \), as these are not in \( A \) but in \( U \). It's important to see that complement expresses 'absence' but within the confines of the Universal Set.

• The Complement helps frame problems involving "what is not" in mathematical terms, being crucial in logic, probability, and set operations.
• It raises intriguing queries, like when considering the complement of the Universal Set, which is, logically, the Null Set, since nothing exists outside of \( U \).