Lecture 05: The Language of Sets

The language of sets is foundational in mathematics, providing a structured way to define, organize, and relate collections of objects, often called "elements." Sets are typically represented by curly brackets, with elements listed inside, such as A={1,2,3}. This simple notation allows mathematicians to categorize objects, express relationships, and analyze data in a clear, systematic way.

Set operations like union (∪), intersection (∩) and complement enable the combination and comparison of sets, revealing common elements or differences. For example, the union of two sets A and B, written as A ∪ B contains all elements in either A or B. The intersection A ∩ B represents elements shared by both.

Sets also form the basis for defining more complex structures, like functions, probability, and logic. Concepts such as subsets and proper subsets help establish hierarchies within groups, while Venn diagrams visually represent these relationships, making abstract ideas more intuitive.