Empty set

Definition

The empty set, denoted as $ ext{∅}$ or `{}`, is a unique set that contains no elements. It is a fundamental concept in set theory and serves as the identity element for set union and the absorbing element for set intersection. Understanding the empty set is essential for grasping the behavior of set operations and the definition of functions involving sets.

5 Must Know Facts For Your Next Test

1. The empty set is a subset of every set, including itself, which means $ ext{∅} ext{ } ext{⊆} ext{ } A$ for any set A.
2. In terms of cardinality, the empty set has a cardinality of zero, which can be expressed as $|∅| = 0$.
3. The union of any set A with the empty set results in A itself, expressed as $A ext{ } ext{∪} ext{ } ∅ = A$.
4. The intersection of any set A with the empty set results in the empty set, shown as $A ext{ } ext{∩} ext{ } ∅ = ∅$.
5. The concept of an empty set helps define functions on sets, especially when considering cases where the function may not have an output for certain inputs.

Review Questions

• How does the empty set behave in relation to other sets when considering operations such as union and intersection?
  • The empty set has unique properties when performing operations with other sets. Specifically, when you take the union of any set A with the empty set, it simply returns A, as $ ext{A ∪ ∅ = A}$. Conversely, when you intersect any set A with the empty set, you get the empty set itself, represented as $ ext{A ∩ ∅ = ∅}$. These operations illustrate the role of the empty set as both an identity element for union and an absorbing element for intersection.
• Explain how understanding the empty set can aid in defining functions on sets.
  • Understanding the empty set is crucial for defining functions on sets because it allows us to handle cases where there may not be any outputs. For example, if we define a function that maps elements from one set to another and there are no elements in the domain (the input set), then this function would map to the empty set. This shows how important it is to consider situations involving the empty set when working with functions to ensure that definitions remain consistent and mathematically valid.
• Evaluate the implications of including the empty set within discussions about subsets and cardinality in mathematical contexts.
  • Including the empty set in discussions about subsets and cardinality highlights its foundational role in mathematics. Since every set includes the empty set as a subset, it serves as a baseline for comparison. Additionally, its cardinality being zero emphasizes that there is always at least one subset (the empty set) even in sets with no elements. This concept underlines how mathematical structures are built from foundational elements like the empty set, allowing for more complex relationships and functions to emerge within larger sets.

Related terms

set: A collection of distinct objects, considered as an object in its own right.

subset: A set A is a subset of a set B if all elements of A are also elements of B.

cardinality: The cardinality of a set refers to the number of elements in that set.