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

More documents

Recommendations

Info

Karnaugh maps: The area of a K-map rectangle is determined by the number of variables (n) and how many (k) are used in a Boolean expression: 2 n4k . Common arrangements are • 2 variables: 2$2, • 3 variables: 4$2, and • 4 variables: 4$4. Each variable contributes two possibilities to each possibility of every other variable in the system. K-maps are organized so that all the possibilities of the system are arranged in a grid form and between two adjacent boxes only one variable can change value. Each square in a K-map corresponds to a minterm. Cover the ones on the map by rectangule that contain a number of boxes equal to a power of 2 (e.g., 4 boxes in a line, 4 boxes in a square, 8 boxes in a rectangle, etc.). Once the ones are covered, a term of a sum of products is produced by finding the variables that do not change throughout the entire covering, and taking a 1 to mean that variable and a 0 as the complement of that variable. Doing this for every covering produces a matching function. 215 Given a Boolean function f with inputs x 1 , …, x n , make a table with all possible inputs and outputs. Then create a K-map with the variables on the left and top sides of the rectangle. Look for 1’s. The rectangle is a torus, so look for wrap arounds, too. Example: f: B 4 %B with a corresponding K-map of x 1 , x 2 00 01 11 10 00 0 0 1 1 x 3 , 01 0 0 1 1 x 4 11 0 0 0 1 10 0 1 1 1 The K-map is colored to try to find patterns in the Boolean expression that can be simplified. It is quite common to eliminate some of the Boolean variables using this approach. Use high quality software if you use the K-map approach. 216
Definition: An implicant is sum term or product term of one or more minterms in a sum of products. A prime implicant of a function is an implicant that cannot be covered by a more reduced (i.e., one with fewer literals) implicant. Note: Suppose f is a Boolean function and P is a product term. Then P is an implicant of f if f takes the value 1 whenever P takes the value 1. This is sometimes written as P ; f in the natural ordering of the Boolean algebra. Quine-McCluskey: This algorithm has two steps: 1. Find all prime implicants of the function. 2. Use those prime implicants in a prime implicant chart to find the essential prime implicants of the function as well as other prime implicants that are necessary to cover the function. The algorithm constructs a table and then simplifies the table. The method leads to computer implementations for large numbers of variables. Use high quality software if you use the Quine-McCluskey approach. 217
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 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: Examples: Most circuits are of the
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?