Association Rules $($AR$)$ and Collaborative Filtering $($CF$)$ are two distinct methodologies used for extracting insights and making recommendations from large datasets. Understanding the differences between these methods is essential for researchers and practitioners to effectively employ them in various applications.

How would you define Association Rules and Collaborative Filtering methods in the context of data mining? Can you elaborate on the underlying principles and methodologies employed by each approach in extracting patterns and making recommendations?

Discuss the differences in data representation and processing between Association Rules and Collaborative Filtering techniques. How do these methods handle input data, and what are the implications for scalability and computational complexity?

Reflect on the strengths and limitations of Association Rules and Collaborative Filtering methodologies. What are the advantages and disadvantages of each method in terms of interpretability, scalability, and performance? How do these factors influence the selection of an appropriate method for a given data mining task?