· Compound Statements: Demonstrate understanding of the logical connectives conjunction, disjunction and negation by applying logical equivalence laws to reduce complex expressions.
· Quantified Statements: Recognize valid argument forms including the rule of universal instantiation, universal modus ponens and universal modus tollens and be able to correctly identify converse and inverse errors.
· Compound Quantified Statements: Write the negation of compound quantified statements.
· Use the division algorithm to prove parity of the integers.
· Solve problems using integer division and modulo arithmetic.
· Recognize instances of and solve problems using the sum of the first n integers and the sum of a geometric sequence.
· Use logarithms, floor, and ceiling notation in calculations.
· Demonstrate proficiency in matrix arithmetic.
· Demonstrate proficiency with summation and product notation including transformation by change of variable.
· Write simple proofs of theorems relating to elementary number theory utilizing several of the following methods: Direct Proof, Disproof by Counter Example, Indirect Arguments using either Contraposition or Contradiction, and Mathematical Induction.
· Use Mathematical Induction to prove the validity of an explicit formula for a recurrence relation.
· Know the definitions of equality, union, intersection, set difference, and complement of sets, empty set, power set, set partition, and Cartesian products.
· Distinguish between "element of" and "subset of".
· Use logical equivalencies to simplify complex set statements.
· Incorporate an axiomatic basis of Boolean Algebra, including DeMorgan's Theorem in problem solving.
· Utilize Canonical Forms: Product Form, Sum Form, Minterms, and Maxterms.
· Demonstrate familiarity with design aids: Karnaugh maps and the Quine-McClusky method.
· Implement logic circuits using AND/OR and NAND/NOR circuits..
· Demonstrate an understanding of recursion using prototypical examples: Towers of Hanoi, Fibonacci numbers, and factorials.
· Solve recurrence relations iteratively.
· Find an explicit formula for second order linear homogeneous recurrence relations given the characteristic equation, single and double root theorems.
· Know basic definitions of multigraph, simple graph, digraph, bipartite graphs, walks, paths, circuits, subgraph, connected graph, and graph components.
· Represent graphs using matrices. Identify basic graph properties in matrices: n-walks, degrees of vertices, in-degree, out-degree, and connectedness.
· Determine isomorphism for small graphs or use isomorphic invariants to show two graphs are not isomorphic.
· Identify Eulerian graphs.
· Be familiar with basic properties of trees, rooted trees, binary trees, full binary trees and m-ary trees.
· Determine the feasibility of basic graphs given properties such as order, size, cardinality of internal or terminal vertices, and score sequence.
· Use either Kruskal's or Prim's algorithm to find a minimum spanning tree.
· Define relations on sets, binary relations, and m-ary relations.
· Identify Reflexive, irreflexive, symmetric, asymmetric, and transitive relations using digraphs, adjacency matrices, and or set elements as applicable.
· Determine whether relations are equivalence or partial order relations.
· Identify equivalence classes of a relation defined on a finite set.
· Demonstrate understanding of sample space.
· Count elements of lists, sublists, and one-dimensional arrays.
· Distinction between finite and infinite sets
· Distinction between countable and uncountable sets
· Use the multiplication and addition rules.
· Use the formulae for permutations and combinations.
· Demonstrate proficiency using the inclusion/exclusion rule.
· Use the binomial theorem and Pascal's triangle.