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

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The textbook has a large number of set identities in a table. Identity Law(s) A2/ = A, A3U = A Identity A2U = U, A3/ = / Domination A2A = A, A3A = A Idempotent A = A Complementation A2B = B2A, A3B = B3A Commutative A2(B2C) = (A2B)2C, A3 (B3C) = (A3B) 3C Associative A3 (B2C) = (A3B) 2 (A3C) Distributive A2(B3C) = (A2B) 3 (A2C) A!B = A"B, A!B = A"B DeMorgan A2 (A3B) = A, A3 (A2B) = A Absorption A!A = U, A"A = # Complement Many of these are simple to prove from very basic laws. 31 Definition: A function f:A%B maps a set A to a set B, denoted f(a) = b for a,A and b,B, where the mapping (or transformation) is unique. Definition: If f:A%B, then • If (b,B )a,A (f(a) = b), then f is a surjective function or onto. • If A=B and f(a) = f(b) implies a = b, then f is one-to-one (1-1) or injective. • A function f is a bijection or a one-to-one correspondence if it is 1-1 and onto. Definition: Let f:A%B. A is the domain of f. The minimal set B such that f:A%B is onto is the image of f. Definitions: Some compound functions include n n • (! f i i )(a) = ! f (a) i=1 i . We can substitute + if we expand the summation. n n • (! f i=1 i )(a)=! f (a) i=1 i . We can substitute * if we expand the product. 32
Definition: The composition of n functions f i : A i %A i+1 is defined by (f 1 °f 2 °…°f n )(a) = f 1 (f 2 (…(f n (a)…)), where a,A 1 . Definition: If f: A%B, then the inverse of f, denoted f -1 : B%A exists if and only if (b,B )a,A (f(a) = b ! f -1 (b) = a). Examples: • Let A = [0,1] 1 R, B = [0,2] 1 R. o f(a) = a 2 and g(a) = a+1. Then f+g: A%B and f*g: A%B. o f(a) = 2*a and g(a) = a-1. Then neither f+g: A%B nor f*g: A%B. • Let B = A = [0,1] 1 R. o f(a) = a 2 and g(a) = 1-a. Then f+g: A%A and f*g: A%A. Both compound functions are bijections. o f(a) = a 3 and g(a) = a 1/3 . Then g°f(a): A%A is a bijection. • Let A = [-1, 1] and B=[0, 1]. Then o f(a) = a 3 and g(a) = {x>0 | x= a 1/3 }. Then g°f(a): A%B is onto. 33 Definition: The graph of a function f is {(a,f(a)) | a,A}. Example: A = {0, 1, 2, 3, 4, 5} and f(a) = a 2 . Then (a) graph(f,A) (b) an approximation to graph(f,[0,5]) 34
Page 1 and 2: Discrete Mathematics University of
Page 3 and 4: Definition: A proposition is a stat
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: Definition: n sets A i are disjoint
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 •
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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
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Examples: Most circuits are of the
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Definition: An implicant is sum ter
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Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

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