# CS 303 Discrete Structures

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## Catalog Description

This course is an introduction to discrete mathematics for computer science. The course covers the basic topics from set theory (including functions and relations), logic, number theory, counting, graph theory, and discrete probability. It involves a detailed study of proof techniques. Prerequisite - C or better in CS 201.

## Prerequisites

• Willingness to work hard.

## Standard Content

### Course Outline

• Logic: proof tables, existential quantification, universal quantification, proposition, propositional function, De Morgan’s law (R. Ch. 1 - “R.” refers to the Rosen book, see below.)
• Basic set theory: sets, intersection, union, complement, Venn diagram, inclusion-exclusion principle (R. Ch. 2)
• Proof techniques: direct proof, contrapositive proof, proof by contradiction, weak induction, strong induction (P. Part 2 and R. Ch. 5 - “P.” stands for the “Book of Proof”, see below.)
• Counting: permutations, combinations, binomial theorem, identities (e.g. {Pascal’s identity) pigeonhole principle and applications, double counting (R. Ch. 6)
• Discrete probability: sample space, events experiments, outcomes, distributions, conditional probability, random variables, independence, expectation, linearity of expectation (R. Ch. 7)
• Optional topics: Monty Hall problem, 100 prisoners problem, probabilistic method to prove lower bound on Ramsey numbers
• Number theory: modular arithmetic division algorithm, modular exponentiation, primes, sieve or Eratosthenes, gcd, lcm, Euclidean algorithm, Bezout’s theorem, inverses, fundamental theorem of arithmetic, solving linear congruences, affine cipher, RSA (R. Ch. 4)
• Graph theory: basic graph terminology, undirected graphs, bipartite graphs, degree of a vertex, handshaking theorem, matchings, Hall’s marriage theorem (optional) (R. Ch. 10)

### Learning Outcomes

• Capable of constructing rigorous mathematical proofs using direct proof, proof by contradiction, induction, etc.

### Important Assignments and/or Exam Questions

• What is the decryption function for an aﬃne cipher if the encryption function is c = (15p + 13) mod 26?
• Show that if 7 integers are selected from {1,2,3, ..., 12}, then there must be a pair of these integers with a sum equal to 13. What if only 6 integers are selected rather than 7? Can you still conclude that there must be a pair of these integers with a sum equal to 13?
• A k-regular graph G is one such that deg(v) = k for all v ∈ G. Prove that if G is a k-regular bipartite graph with k > 0 and the bipartition of G is A and B, then |A| = |B|.
• What is the probability of these events when we randomly select a permutation of 1,2,3,4?
• 1 precedes 4
• 4 precedes 1
• 4 precedes 1 and 4 precedes 2
• 4 precedes 1, 4 precedes 2, and 4 precedes 3
• 4 precedes 3 and 2 precedes 1.
• Let Xn be the random variable that equals the number of tails minus the number of heads when n fair coins are ﬂipped. Compute the expected value of Xn. (Hint: Deﬁne a set of a random variables and express Xn in terms of these random variables. Then use linearity of expectation. Explain clearly everything you do.)
• Prove by induction that for any positive integer n, the inequality (1 + x)n ≥ 1 + nx holds for all x ∈ R with x > −1.
• Suppose that the domain of the propositional function P(x) consists of the integers −1, 0, and 1. Write out each of these propositions using disjunctions, conjunctions, and negations: (a) ∃xP(x), (b) ∃x¬P(x), (c) ¬∃xP(x), (d) ¬∀xP(x).