1. Probability vs Fuzziness
2. Binary World
2.1. Indicator function
3. Fuzzy World
3.1. Membership function
4. Set theory
4.1. Crisp Set Theory (Classical/Binary)
4.1.1. Discrete vs Continuous sets
4.1.2. Crisp set operations
4.1.3. Defining indicator functions for set operations
4.2. Fuzzy Set Theory
4.2.1. Notation - Discrete vs Continuous sets
4.2.2. Fuzzy set operations
5. Cartesian products and relations
5.1. Cartesian products
5.1.1. crisp set
5.1.2. fuzzy set
5.2. Relations (subsets of cartesian products)
5.2.1. crisp relations
5.2.1.1. cardinality
5.2.1.2. equivalence relation
5.2.1.2.1. reflexive
5.2.1.2.2. symmetric
5.2.1.2.3. transitive
5.2.1.2.4. equivalence classes
5.2.1.3. composition of relations
5.2.1.4. tolerance relations
5.2.2. fuzzy relations
5.2.2.1. fuzzy equivalence relations
6. Logic systems
6.1. scheme for a logical system
6.2. fuzzy systems
6.2.1. input
6.2.2. fuzzification
6.2.3. inference engine
6.2.4. defuzzification
6.2.4.1. lambda cuts
6.2.5. feedback
6.2.6. output
7. Applications of fuzzy logic
7.1. Clustering - unsupervised learning
7.1.1. clustering in classical (crisp) sense
7.1.1.1. properties of clustering
7.1.2. Optimal Clustering: -Minimizing intra-cluster variance -Maximizing inter-cluster separation this means a minimal objective function
7.1.2.1. k-mean clustering algorithm
7.1.2.1.1. initializing the membership matrix U at iteration r=0. (initial assignments of points to clusters)
7.1.2.1.2. At each iteration r, you compute the centroids or cluster centers Vj(r) of the clusters, where j represents the cluster index. (mean of points in each cluster)
7.1.2.1.3. Update the assignments U(r) (assign points to nearest centroids).
7.1.2.1.4. Evaluate if the algorithm has converged (centroids or assignments stop changing).
7.1.3. challenges in crisp k-means
7.1.4. fuzzy k-means
7.1.5. fuzzy clustering problem
7.1.6. objective functions
7.1.7. fuzzy k-means algorithm
7.1.8. similarity between mutual data points
7.1.9. fuzzy relation between clusters