Zermelo-Fraenkel set theory, often abbreviated as ZF, is a foundational system for mathematics that describes how sets, or collections of objects, can be formed and manipulated. It provides a formal framework to understand the relationships between different sets, ensuring that they are well-defined and free from contradictions. This theory is essential for modern mathematics, as it underpins many mathematical concepts and structures. One of the key features of ZF is the use of axioms, which are basic rules that govern how sets behave. For example, the Axiom of Extensionality states that two sets are equal if they contain the same elements. Together with the Axiom of Choice, which is often added to form ZFC, Zermelo-Fraenkel set theory