Discrete Mathematics (Spring 2013)
1. Propositions. Logical Operations: negation, conjunction, disjunction, implication, exclusive or,
biconditional. Truth Tables. Bitwise logical operations. Tautology and a contradiction. Prove that (p → (q
→ r)) → ((p → q) → (p → r)) is a tautology without using truth tables. Propositional Equivalences. Tables
of Logical Equivalences.
2. Truth Tables of basic logical operations. The number of all compound propositions consisting of n
elementary compositions. Prove this by mathematical induction. Construct the truth table for the proposition
(𝑝 → 𝑞) ∧ (𝑞⨁𝑟) ∧ (𝑞 → (𝑝 ∨ 𝑞)).
3. Elementary conjunction and elementary disjunctions. Conjunctive Normal Form and Disjunctive Normal
Forms of formulae of Proposition Logic. Prove that for any proposition there is an equivalent proposition
which is either DNF, or CNF.
4. Functionally complete collection of logical operators. Show that both {¬, &, V} and {¬, &} form a
functionally complete collection of logical operators. Express all basic logical operations by these
operations.
5. Functionally complete collection of logical operators. Show that both {¬, V} and {¬, →} form a
functionally complete collection of logical operators. Express all basic logical operations by these
operations.
6. Functionally complete collection of logical operators. Show that both { | } and { ↓ } form a functionally
complete collection of logical operators. Express all basic logical operations by these operations.
7. Functionally complete collection of logical operators. Show that {+, ∙, 1} forms a functionally complete
collection of logical operators. Express all basic logical operations by these operations.
8. Language of Predicate Logic. Definition of a term and of a formula of First Order Logic. Examples.
9. Quantifiers. Existential Quantification, Universal quantification. Free and bound variables. Examples. Table
of Negating Quantifiers. Prenex Normal Form of Formulae.
Show that  x(P(x)  Q(x)) and  x P(x)   x Q(x) are logically equivalent.
Show that  x(P(x)  Q(x)) and  x P(x)   x Q(x) are not logically equivalent.
Show that  x(P(x)  Q(x)) and  x P(x)   x Q (x) are logically equivalent.
Show that  x(P(x)  Q(x)) and  x P(x)   x Q (x) are not logically equivalent.
10. Set. Subset. Equality of two sets. The Power Set. The cardinality of the power set (prove this formula for
finite sets by mathematical induction). Cartesian Products. The cardinality of the Cartesian product of two
sets. Computer representation of sets.
11. Set Operations. Venn Diagrams of all essential set operations. Set Identities. The cardinality of the union of
two sets. Prove by Venn Diagram de Morgan laws.
12. Function. Domain. Codomain (range). Image and pre-image of an element and of a set. Injective and
surjective functions. Bijection. Examples. Graph of a function. Floor and Ceiling functions. Inverse
function. Composition of functions. Prove that the composition of injective (surjective, bijective) functions
is an injective (surjective, bijective) function.
13. Cardinality of a set. Definitions of |A| = |B| and |A| ≤ |B|. Countable sets. Show that the set of integer and the
set of rational numbers are countable.
14. Prove that |P(A)| = 2n for any set A, where n = |A|. Prove that |P(A)| > |A| (Cantor’s Theorem).
15. Show that the set of reals is uncountable. Show that there is a bijection between the sets of all even natural
numbers and the set of all integers.
16. Show that the image of the union of two sets under a function is equal to the union of the images of these
sets.
Show that the image of the intersection of two sets under a function is a subset of the intersection of the
images of these sets.
17. Show that the pre-image of the union of two sets is equal to the union of pre-images of these two sets.
Show that the pre-image of the intersection of two sets is equal to the intersection of pre-images of these two
sets.
Show that the complement of the pre-image of a set is equal to the pre-image of the complement of this set.
18. Let f be a bijection. Show that the image of the intersection of two sets under f is equal to the intersection of
the images of these sets under f.
Show that the complement of the image under f of a set is equal to the image under f of the complement of
this set.
19. Rules of inference. The six basic rules of inference. Vacuous, trivial, indirect proof and proof by
contradiction. Mathematical Induction. Well-ordering property. Basic and inductive steps. Why
mathematical induction is valid. Second principle of mathematical induction. Show that
12 + 22 + … + n2 = n(n+1)(2n+1)/6.
20. Basic counting principles. The sum rule. The product rule. Principle of inclusion-exclusion. Number of all
functions from a set A to a set B. The Pigeonhole principle. The Generalized Pigeonhole principle. Show
that there are either three mutual friends or three mutual enemies in a group of six people, such that each
pair of individual of the group consists of two friends or two enemies.
21. The Pigeonhole principle. Show that among any n + 1 positive integers not exceeding 2n must be an integer
that divides one of the other integers.
22. The Pigeonhole principle. Show that every sequence of n2 + 1 distinct real numbers contains a subsequence
of length n + 1 that is either strictly increasing or strictly decreasing.
23. r-permutations and r-combinations. Prove theorems on P(n,k), C(n,k).
24. Binomial coefficient. The binomial theorem. Expand brackets in (a1+a2+…+ak)n.
25. Pascal Identity (give a combinatorial proof). Give a combinatorial proof of C(2, 2n) = 2C(2, n) + n2.
26. Vandermonde’s Identity (with a proof).
27. Show in two different ways that C(n, 0) + C(n, 1) + … + C(n, n) = 2n.
Show that C(n, 0) – C(n, 1) + … + (–1)kC(n, k) + … + (–1)nC(n, n) = 0.
28. Generalized permutation and combinations. Permutations and combinations with repetition. Permutations of
sets with indistinguishable objects, distributing objects into boxes.
29. Recurrence relations. Solution of a recurrence relation. The tower of Hanoi. Codeword enumeration.
30. Solving recurrence relations. Linear homogeneous recurrence relation of degree k. Characteristic equation.
Solving recurrence relations of the second degree.
31. Solving recurrence relations of degree k. Solving linear inhomogeneous recurrence relations.
32. Relations. Properties of relations (reflexivity, symmetry, etc). N-ary relations, and their domain and degree.
Projections.
33. The reflexive closures. Prove that the intersection of reflexive relations is reflexive. Describe what is the
reflexive closure on the digraph and the matrix representing a relation. Prove that the reflexive closure of a
relation R is equal to the intersection of all reflexive relations containing R.
34. The symmetric closures. Prove that the intersection of symmetric relations is symmetric. Describe what is
the symmetric closure on the digraph and the matrix representing a relation. Prove that the symmetric
closure of a relation R is equal to the intersection of all symmetric relations containing R.
35. The transitive closures. Prove that the intersection of transitive relations is transitive. Describe what is the
transitive closure on the digraph representing a relation. Prove that the transitive closure of a relation R is
equal to the intersection of all transitive relations containing R.
36. Give a definition of the transitive closure. Prove that if a relation R is defined on a set with n elements, then
if there is a path from a to b, then there is a path from a to b of length at most n. Give a matrix
representation of the transitive closure of a relation defined on an n-elements set.
37. Warshall’s algorithm of constructing the transitive closure. Calculating the numbers of operations of
Warshall’s algorithm and of the standard algorithm.
38. Combining relations. Composite of two relations. The powers of a relation. Matrices representing relations
and their composition. The number of ways of length k from one element to another one. Representing
relations using matrices and digraphs. Digraph. Path in directed graphs.
39. Path in directed graphs. The shortest path problem. Dijkstra’s algorithm.
40. Equivalence relations. Equivalence classes. Partitions. Prove properties of equivalence classes. Prove that
for a given partition it is possible to construct an equivalence relation. Refinement.
41. Partial orderings. Comparable and incomparable elements. Linear ordering. Lexicographic order. Chains.
Show that if there is a unique minimal element in a finite poset, then this element is the least.
42. Hasse diagrams. Maximal and minimal, greatest and least elements. Bounds. Lattices. Show that each finite
poset has a minimal element.
43. Show that every totally ordered set is a lattice. Show that every finite lattice has both the least and the
greatest elements.
44. Types of graphs. Basic terminology. Theorem on the sum of the degrees for undirected graphs with a proof.
45. Terminology for directed graphs. Theorem on the sum of the degrees for directed graphs with a proof.
Special simple graphs. Bipartite graphs. Connectivity. Connected components.
46. Isomorphism of graphs and its properties. Examples. Example with Peterson’s graph.
47. An Euler circuits and paths. Necessary and sufficient conditions for Euler circuits and paths with a proof.
48. Hamilton paths and circuits. For which values of n and m the complete bipartite graph Kn,m have a Hamilton
circuit?
49. Planar Graphs. Is K3,3 planar (with a proof)? Homeomorphism. Prove that Peterson’s graph contains a
subgraph which is homeomorphic to K3,3.
50. Planar Graphs. Euler formula and its corollary (with a proof). Is K5 planar? Kuratowski's theorem.
51. The chromatic number of a graph. A graph which is dual for a map. The chromatic number of basic graphs
and of a planar graph.
52. Tree, tree traversal (infix, prefix, postfix), polish notation.
53. Turing Machine. Write Turing Machines for calculating the function f(x) = x+3 and for the function g(x) = 2.
54. Algorithms. Gödel enumeration. Church thesis. Prove that for the halting problem there is no algorithmic
solution (problem that a Turing Machine will terminate).