Lecture
This lecture focuses on the properties of injective functions, particularly in the context of set theory. The instructor begins by presenting a key example and defining the sets involved. The proof of injectivity is demonstrated in two parts, starting with the assumption that the function is injective and showing the necessary inclusions. The instructor illustrates the concept using diagrams to clarify the relationships between the sets. The lecture then transitions to the reverse direction, where the instructor assumes the inclusion property holds for all subsets and aims to prove that the function is injective. This involves selecting distinct elements and demonstrating that their images under the function are also distinct. The instructor uses visual aids, including GeoGebra files, to provide concrete examples of functions and their graphs, emphasizing the importance of understanding the behavior of functions in different scenarios. The session concludes with a brief Q&A segment, allowing participants to clarify their understanding of the material presented.