Topics for the Final Exam:
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- All material from 8/25/15 until 12/4/15 (inclusive)
- This includes material covered in the lectures and labs, and all homeworks and quizzes



From Test 1:
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Logic: 
(1.1-1.6; 12.1, 12.2)
	- Proving that logical formulas are equivalent: both by truth tables and logical equivalences
        - Know basic rules of equivalence
	- De Morgan's law for negation
	- Quantifiers, negation
	- Translating from English to logic, and logic to English
	- Boolean functions (12.1, 12.2)
	- Know and use rules of inference

Proofs:
(1.7, 1.8)
	- Direct proof
	- Contradiction, contrapositive
	- Proof by cases
	- Prove a collection of statements are equivalent
	- Know how to prove statements rigorously
	- Know how to disprove statements (e.g., counterexample)

Sets:
(2.1, 2.2, 2.5)
	- Set operations: union, intersection, complement, set minus
        - Cartesian product (tuples)
        - Power set
	- Set relations: element of, subset of, equality
	- Know how to prove two sets are the same
	- De Morgan's law for sets
	- Cardinality 

Functions:
(2.3)
	- Definition of a function: domain, codomain, and the mapping
	- Injection, Surjection, Bijection - you should be able to prove or disprove a function is any of these, and give examples
	- Function composition, inverse

Sequences and Series (Summations):
(2.4)
	- Find formulas that generate a sequence by looking for a pattern
	- Know the geometric series and arithmetic series 

Induction:
(5.1, 5.2)
        - Weak induction, strong induction
	- Be able to perform inductive proofs
	- Know the difference between weak and strong induction


New material:
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Recursive Definitions:
(5.3, 8.1, 8.2)
	- Know how to recursively define: a set, a function, a sequence, and know the difference between these three
	- Prove recursive definitions are correct using induction
	- Fibonacci numbers, towers of Hanoi (8.1)
        - Solving linear recurrence relations (8.2)

Number Theory:
(4.1, 4.3, 4.4, 4.6)
	- Definition of mod, divides
	- GCD, LCM, relatively prime
	- Definition of prime, prime factorization
	- Euclidean Algorithm
	- Modular Inverses: when do they exist, and how do we find them
	- Solve linear congruences
	- RSA: understand why it works, and how to use the algorithm with reasonable numbers

Combinatorics:
(6.1-6.5, 8.5)
	- Combinations, permutations, repetition, no repetition: definitions, when to use them, formulas
        - Addition, multiplication principle
	- Inclusion-exclusion principle (8.5)
        - Pigeonhole principle
	- Word problems
	- Counting functions from one set to another
	- Binomial coefficients and the binomial theorem

Probability:
(7.1-7.4)
	- Definition: a sample space, probability distribution over that sample space, random variables
	- Expected value and variance, rules about linearity
	- The basic distributions covered in class
	- Conditional probability
	- Independence of events, random variables
	- Finding probabilities by counting
	- Bayes' theorem

Relations:
(9.1, 9.5)
        - Binary relations
	- Properties of relations (reflexive, symmetric, anti-symmetric, transitive)
	- Equivalence relations 
	- Equivalence classes, congruence classes

Graphs:
(10.1-10.4, 10.7, 11.1)

	- Definition of undirected and directed graphs; G=(V,E) with E being unordered or ordered vertex pairs
	- Basic definitions: adjacent, degree, paths, connected
	- Storage: Adjacency lists, adjacency matrix; handshaking lemma
	- Trees, rooted trees
	- Induction on trees and graphs
	- Planar graphs: definition, non-planar graph examples, Euler's formula