Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

More documents

Recommendations

Info

Course Outline 1. Logic Principles 2. Sets, Functions, Sequences, and Sums 3. Algorithms, Integers, and Matrices 4. Induction and Recursion 5. Simple Counting Principles 6. Discrete Probability 7. Advanced Counting Principles 8. Relations 9. Graphs 10. Trees 11. Boolean Algebra 12. Modeling Computation 3 Logic Principles Basic values: T or F representing true or false, respectively. In a computer T an F may be represented by 1 or 0 bits. Basic items: • Propositions o Logic and Equivalences • Truth tables • Predicates • Quantifiers • Rules of Inference • Proofs o Concrete, outlines, hand waving, and false 4
Definition: A proposition is a statement of a true or false fact (but not both). Examples: • 2+2 = 4 is a proposition because this is a fact. • x+1 = 2 is not a proposition unless a specific value of x is stated. Definition: The negation of a proposition p, denoted by ¬p and pronounced not p, means that, “it is not the case that p.” The truth values for ¬p are the opposite for p. Examples: • p: Today is Thursay, ¬p: Today is not Thursday. • p: At least a foot of snow falls in Boulder on Fridays. ¬p: Less than a foot of snow falls in Boulder on Fridays. 5 Definition: The conjunction of propositions p and q, denoted p!q, is true if both p and q are true, otherwise false. Definition: The disjunction of propositions p and q, denoted p"q, is true if either p or q is true, otherwise false. Definition: The exclusive or of propositions p and q, denoted p#q, is true if only one of p and q is true, otherwise false. Truth tables: p ¬p q p!q p"q p#q T F T T T F T * F * F F T T F * T * T F T T F T F F F F * The truth table for p and ¬p is really a 2$2 table. 6
Page 1: Discrete Mathematics University of
Page 5 and 6: We can compound logical operators t
Page 7 and 8: Theorem: p" (q!r) ' (p"q)!(p"r). Pr
Page 9 and 10: Propositional logic is pretty limit
Page 11 and 12: Rules of Inference are used instead
Page 13 and 14: Sets, Functions, Sequences, and Sum
Page 15 and 16: Definition: n sets A i are disjoint
Page 17 and 18: Definition: The composition of n fu
Page 19 and 20: Some other common summations with c
Page 21 and 22: Notation: Timing, as a function of
Page 23 and 24: ! $ procedure binary_search(x, " a
Page 25 and 26: Definition: If a,b,Z and a-0, we sa
Page 27 and 28: Theorem: There are infinitely many
Page 29 and 30: Algorithm: Addition of integers pro
Page 31 and 32: Euclidean Algorithm: Compute gcd(a,
Page 33 and 34: Sun Tzu’s Puzzle: The a k ,{2, 1,
Page 35 and 36: • Sometimes P(k) is for a (possib
Page 37 and 38: Example: Every postage amount < $.1
Page 39 and 40: Definition: A recursive algorithm s
Page 41 and 42: Merge sort is a balanced binary tre
Page 43 and 44: Examples: • Consider 3 students i
Page 45 and 46: Theorem: The number of r-combinatio
Page 47 and 48: If we allow repetitions in the perm
Page 49 and 50: Algorithm: Generating the next r-co
Page 51 and 52: Theorem: Let E and F be events in a
Page 53 and 54:
Note: For bit strings of length 4,
Page 55 and 56:
Theorem: If X i , 1;i;n, are random
Page 57 and 58:
Definition: The random variables X
Page 59 and 60:
Example: We have 2 boxes. The first
Page 61 and 62:
Let E be the event that an incoming
Page 63 and 64:
Advanced Counting Principles Defini
Page 65 and 66:
Examples: Typical ones include •
Page 67 and 68:
Now comes the second case for n = 2
Page 69 and 70:
Theorem: Assume {b i },{c i },R. Su
Page 71 and 72:
Example (funny integer multiplicati
Page 73 and 74:
• In step 4, we have to take care
Page 75 and 76:
Definition: The extended binomial c
Page 77 and 78:
Example: a n = 8a n41 + 10 n41 with
Page 79 and 80:
Relations Definition: A relation on
Page 81 and 82:
Representation: The relation R from
Page 83 and 84:
! # 0 0 0 0 # • M R4 = # 1 0 0 0
Page 85 and 86:
Definition: A partition of a set S
Page 87 and 88:
Examples of Simple Graphs: • A co
Page 89 and 90:
Examples: v 1 v 2 • and v 4 v 3 v
Page 91 and 92:
Theorem: Let G = (V,E) be a graph w
Page 93 and 94:
Definition: A coloring of a simple
Page 95 and 96:
Definition: A m-ary tree is a roote
Page 97 and 98:
Definition: A decision tree is a ro
Page 99 and 100:
Note: Game trees are another highly
Page 101 and 102:
Definition (Postorder Traversal): L
Page 103 and 104:
Definition: Let G = (V,E) be a simp
Page 105 and 106:
Definition: A minimum spanning tree
Page 107 and 108:
Examples: Most circuits are of the
Page 109:
Definition: An implicant is sum ter
show all

Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Create successful ePaper yourself

Delete template?

Save as template?