Discrete Mathematics for Computer Science

1. Sets. Basic Definitions. Exercises. Operations on Sets. Exercises. Mathematical Introduction. Exercises. Strong Form of Mathematical Introduction. Exercises. The Principle of Inclusion-Exclusion. Exercises. Russell's Paradox. 2. Formal Logic. Introduction to Propositional Logic. Exercises. Truth and Logical Truth. Exercises. Normal Forms. Exercises. Formal Proofs in Propositional Logic. Exercises. First Order Logic. Exercises. 3. Relations. Binary Relations. N-ary Relations. Operations on Binary Relations. Exercises. Special Types of Relations. Exercises. Equivalence Relations. Exercises. Ordering Relations. Exercises. Relational Databases-An Application. Exercises. 4. Functions. Basic Definitions. Exercises. Operations on Functions. Sequences and Subsequences. Exercises. The Pigeon-Hole Principle. Applications. Exercises. Countable and Uncountable Sets. Exercises. Languages and Regular Sets. Exercises. 5. Analysis of Algorithms. Comparing Growth Rates of Functions. Exercises. Complexity of Programs. Exercises. Uncomputability. Exercises. 6. Graph Theory. Introduction to Graph Theory. The Handshaking Problem. Paths and Cycles. Graph Isomorphism. Representation of Graphs. Exercises. Connected Graphs. The Koingsberg Bridge Problem. Exercises. Trees. Spanning Trees. Rooted Trees. Exercises. Directed Graphs. Scheduling a Meeting Facility. Finding a Cycle in a Directed Graph. Priority in Scheduling. Connectivity in Directed Graphs. Exercises. 7. Counting and Combinatorics. Traveling Salesperson Problem. Counting Principles. Set Decomposition Principles. Exercises. Permutations and Combinations. Constructing the k-th Permutation. Exercises. Repetitions and Identities. Combinatorial Identities. Pascal's Triangle. Multinomials. TSP Revisited. Exercises. 8. Discrete Probability. Ideas of Chance in Computer Science. Exercises. Cross Product Sample Spaces. Exercises. Independent Events and Conditional Probability. Exercises. Discrete Random Variables. Exercises. 9. Recurrence Relations. The Tower of Hanoi Problem. First Order Recurrence Relations. Exercises. Second Order Recurrence Relations. Exercises. Binary Search. Merge Sort. Multiplication of n-bit Numbers. Divide and Conquer Recurrences. Exercises. 10. Geometric Series and Partial Fractions. Exercises.