CS 205: Introduction to Discrete Structures

Rutgers University
Spring 2005
Michael L. Littman
Lynn Chan, TA

Time: MW 1:10-2:30
Place: Hardenbergh Hall, Room B2
Recitation: Tuesday 1:25-2:20
Place: Hardenbergh Hall, Room A3
Semester: Spring 2005

Michael's office hours: Hill 409, by appointment.
Lynn's office hours: Hill 205, Tue 3pm-4.30pm, Thu 4-4.30pm.


Textbook

Discrete Mathematics and its Applications
Rosen, Fifth Edition, McGraw Hill, 2003.

Links

Lectures, 2004

Date Sections Topics Problems
1/19 1.6, 1.7 sets, subsets assigned 1/24, due 1/31
1.6: 4, 6, 8, 10, 12, 14, 16, 18, 22, 24.c
1.7: 2, 4, 10, 20, 22 (justify your answer), 24, 38, 40, 48 (see definition in 47), 50 (see definition after 48, assume the university will buy all the equipment and departments will borrow what they need temporarily)
1/24 1.8 functions mapping set to set, domain, co-domain, range, one-to-one, onto, inverse, composition
1/26 1.1 Boolean, and, or, not, xor, truth tables, implication, contrapositive, converse, inverse
1/31 1.2 translating logic expressions, equivalences, tautology, contradiction, contingency, DeMorgan, distributive, associative assigned 1/31, due 2/7
1.1: 10, 12, 16, 22, 24(d,e), 28(c,f), 30
1.8: 2(b,c) (justify your answer), 6, 12, 14(a,b,if not, give counterexample), 16, 18, 22 (give values for all elements of S), 24 (name the range), 28, 30, 34 (optional), 66.b
2/2 10.1, 10.2 complementation, Boolean expressions (recursive definition), literal, minterm, DNF, CNF
2/7 10.3 (Lynn) combinatorial circuit, and gate, or gate, inverter, half adder, full adder
2/9 1.3 predicate, propositional function, universal quantification, existential quantification, predicate calculus, connection to conjunction and disjunction
2/14 1.4, 1.5 (start) nested quantifiers, negating nested quantifiers, proof, axioms, rules of inference, fallacy, lemma, corollary, conjecture assigned 2/14, due 2/21 (moved to 2/23)
1.1 (optional): 34(b, c; see pg. 14 for bitwise definition), 44, 48 (consistent means "their conjunction is not a contradiction"), 52, 54
1.2 (optional): 4(a), 6, 10 (detailed proofs, see pg. 25 for example), 12, 30, 40
10.1 (optional): 2, 4(b,d), 6(b,d), 26, 32 (detailed proof), 34 (detailed proof)
1.3: 2(a,b), 10, 12, 22
1.7: 6(a,d,e,g, use quantifiers), 12(a,d, use quantifiers)
10.2: 4(c,d, sum of products means DNF), 6, 8 (answer should be in CNF), 12(c,d), 18 (downarrow is "nor"), 20(a,c, justify your answer)
10.3: 2 (give formula), 4 (give formula), 6(c,d), 10, 12 (build the circuit, using the full and half subtractors as components), 16, 18
2/16 1.5 valid arguments, modus ponens, universal/existential instantiation/generalization, direct proof, indirect proof, proof by contradiction, proof by cases, non-constructive existence proof, strategy-stealing argument
2/21 MIDTERM 1
2/23 3.1, 3.2 proof strategies assigned 2/23, due 3/1
1.3: 26, 30, 34, 38, 40, 42 (supply the inference rules), 46 (give counterexample), 50 (see notation in problem 48), 56, 58
1.4: 8, 10, 12(j,k,l), 16(d,e), 22, 30, 38 (say "true" if no counterexample exists), 42, 48
1.5: 12 (name the inference rule of fallacy for each), 14 (evaluate the validity of the reasoning, not the validity of the statements), 16, 18, 20, 28, 36, 38, 42, 48, 50, 52, 54, 56, 60 (supply the inference rule), 64 (optional), 74
2/28 3.3 induction, basis, inductive hypothesis
3/2 3.4 strong induction assigned 3/2, due 3/21
3.1: 2, 4, 6 (hint: from class), 8 (hint: from class), 12 (compare quadratic mean, a, and b), 20, 22, 26, 28, 32 (optional)
3.2: 6(d,e,h), 10(a,e), 20, 30, 38 (optional), 42 (optional)
3.3: 4, 6 (cute one!), 8, 12, 14 3.3: 16, 18, 24, 28, 30, 36, 42 (ask if you need the definition of matrix multiplication), 44, 48, 50, 52, 54, 58 (Hn = 1/1 + 1/2 + ... + 1/n)
3.4: 6, 8, 12 (fn is the nth Fibonacci number), 14 (hint: use a proof by induction and carefully apply the definition of fn), 18, 22, 26 (optional), 28, 30, 36 (use the definition from the solution to problem 35), 42, 44, 46, 48 (see the definition immediately above), 50, 54
3/7 3.4 (more), 2.4, 2.5, 2.6 (samples) number theory, mod, Chinese remainder theorem
3/9 3.5 recursion, iteration, repeated squaring
3/14, 3/16 spring break
3/21 (guest) 7.1, 7.3 relations, reflexive, symmetric, antisymmetric, transitive, directed graph, matrices
3/23 7.4 closure, path, connectivity relation
3/28 MIDTERM 2
4/2 7.5, 7.6 equivalence relation, equivalence classes, partition, partial order, partially ordered set, incomparable elements, totally ordered, maximal/minimal/maximum/minimum element
4/4 11.1 natural language, vocabulary, sentence, empty string, language, phrase-structure grammar, directly derivable, derivable, derivation assigned 4/4, due 4/12
7.1: 4, 8, 20 (see definition before problem 16), 22, 24 (see definition above the problem), 30, 36, 48 (optional), 56
7.3: 8, 14(c,d,e), 20, 24 (list the ordered pairs)
7.4: 2, 10, 14 (optional), 18, 22 (optional), 24 (optional)
7.5: 2, 10 (optional), 24, 28 (optional)
Puzzle: Puzzle 1: Digital Logic.
4/6 11.1, 11.3 parse tree, finite state automaton
4/11 (guest) 11.2 finite state transducer, regular sets, regular languages, NFAs, DFAs
4/13 11.3, 11.4 Kleene's theorem assigned 4/13, due 4/20
7.5: 30, 34 (recall [x,y] includes the endpoints and (x,y) doesn't), 40, 42
7.6: 4, 6, 8 (bar means "divisibility"), 10, 16, 22, 26, 28, 30 (can draw a diagram), 38, 48 (optional), 56
11.1: 2, 6 (give a proof by induction)
4/18 11.4 pumping
4/20 11.5, 2.2 halting problem, Big O (not yet) assigned 4/20, due 4/27
11.1: 8 (show everything generated has this form and everything of this form can be generated), 12, 16, 18, 24
11.3: 2, 8 (use definition of set equality), 12 (give a regular expression for your answer), 22, 24, 26
4/25 2.2, A.1 big-O, big-Omega, big-Theta, logs, exponentials Practice only:
2.2: 2, 6, 8, 12, 14(d,e,f), 18, 22, 26, 36
11.1: 20
11.3: 2, 8.b, 10, 18
4/27 wrap up
5/2 juggling, site-swap notation
5/10 MIDTERM 3 (12pm-1:30pm), FINAL (1:30pm-3pm)