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

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Concepts so far can be extended to Boolean variables and Bit strings. Definition: A bit is a binary digit. Hence, it has two possible values: 0 and 1. Definition: A bit string is a sequence of zero or more bits. The length of a bit string is the number of bits. Definition: The bitwise operators OR, AND, and XOR are defined based on !, ", and #, bit by bit in a bit string. Examples: • 010111 is a bit string of length 6 • 010111 OR 110000 = 110111 • 010111 AND 110000 = 010000 • 010111 XOR 110000 = 100111 7 Definition: The conditional statement is an implication, denoted p%q, and is false when p is true and q is false, otherwise it is true. In this case p is known as a hypothesis (or antecedent or premise) and q is known as the conclusion (or consequence). Definition: The biconditional statement is a bi-implication, denoted p&q, and is true if and only if p and q have the same truth table values. Truth tables: p q p%q p&q T T T T T F F F F T T F F F T T 8
We can compound logical operators to make complicated propositions. In general, using parentheses makes the expressions clearer, even though more symbols are used. However, there is a well defined operator precedence accepted in the field. Lower numbered operators take precedence over higher numbered operators. Examples: • ¬p!q = (¬p) !q • p!q"r = (p!q) "r Operator Precedence ¬ 1 ! 2 " 3 % 4 & 5 9 Definition: A compound proposition that is always true is a tautology. One that is always false is a contradiction. One that is neither is a contingency. Example: p ¬p p!¬p p"¬p T F F T F T F T contigencies contradiction tautology Definition: Compound propositions p and q are logically equivalent if p&q is a tautology and is denoted p'q (sometimes written as p!q instead). 10
Page 1 and 2: Discrete Mathematics University of
Page 3: Definition: A proposition is a stat
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,
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Theorem: If X i , 1;i;n, are random
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Definition: The random variables X
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Example: We have 2 boxes. The first
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Let E be the event that an incoming
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Advanced Counting Principles Defini
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Examples: Typical ones include •
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Now comes the second case for n = 2
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Theorem: Assume {b i },{c i },R. Su
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Example (funny integer multiplicati
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• In step 4, we have to take care
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Definition: The extended binomial c
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Example: a n = 8a n41 + 10 n41 with
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Relations Definition: A relation on
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Representation: The relation R from
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! # 0 0 0 0 # • M R4 = # 1 0 0 0
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Definition: A partition of a set S
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Examples of Simple Graphs: • A co
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Examples: v 1 v 2 • and v 4 v 3 v
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Theorem: Let G = (V,E) be a graph w
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Definition: A coloring of a simple
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Definition: A m-ary tree is a roote
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Definition: A decision tree is a ro
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Note: Game trees are another highly
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Definition (Postorder Traversal): L
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Definition: Let G = (V,E) be a simp
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Definition: A minimum spanning tree
Page 107 and 108:
Examples: Most circuits are of the
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Definition: An implicant is sum ter
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