Final answer:
The empty set is a fundamental concept in mathematics that is considered a subset of every set because it contains no elements, therefore it exists within any set. The existence of the empty set is an axiom in Zermelo-Fraenkel set theory, making it a core component of set theory and logic.
Step-by-step explanation:
The question of whether the empty set exists within a set pertains to set theory in mathematics. The empty set, often denoted by ∅, Ø, or {}, is a unique set that contains no elements. According to the principles of set theory, the empty set is a subset of every set, which by extension means it exists in every set. The reason for this is purely based on the definition of a subset. A set A is considered a subset of set B if every element of A is also an element of B. Since the empty set has no elements, there's nothing that can possibly violate this condition, and therefore, by definition, it is a subset of any set, including itself.
One might also inquire why the empty set does not exist. However, this is a somewhat misguided question as the empty set does indeed exist as an abstract concept within mathematics. It is a fundamental part of set theory and is used in various areas of mathematics and logic. Its existence is assumed as one of the axioms of Zermelo-Fraenkel set theory, which is one of the standard foundations of modern mathematics.