While the final exam will be comprehensive, covering the entire semester,about half of the exam will be on the material from Chapters 7, 8, and 9.
CONCEPTS  METHODS  
Functions and 
Recognizing onto or onetoone functions 
Calculating a recursive function 
The Integers  Congruence: iquo and mod 
Converting between decimal and binary representation Finding prime factorizations Finding gcd and lcm 
Boolean Expressions  Propositions 
Creating truth tables 
Implication  When an implication is true 
Stating the converse 
Defining Sets and Tuples 
Subsets, the empty set, ksubsets  By listing the elements 
Basic Operations on Sets 
DeMorgan's Laws for sets 
Complement, union, intersection, difference 
Quantifiers  Existential 
Creating Venn diagrams 
Multiple quantifiers  Interpreting statements with two or more quantifiers  Negating these statements 
Mathematical Induction 
Predicates as functions 
Coordinating an induction proof 
Counting Methods  Permutations Combinations (sets) Onetoone correspondence 
Multiplication Principle 
A1, A3, A4, A5, A6, A7

E1, E2, E7, E8, E9, E10, E11, E13, E15, E17,
E20, E21, E23, E24, E25 

Relations  Reflexive, Symmetric, Transitive properties 
Drawing Cayley graphs 
A2, A3, A5, A6, A7, A8, A9, A10  E1, E5, E8a, E11, E12, E13, E15, E20  
Graphs  Paths, circuits, trees 
Spanning trees and Kruskal's Algorithm 
A2, A3, A4, A5, A6, A7, A8, A9  E1, E5, E8, E10, E12, E13, E15, E16, E19, E20 