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CSCI 115

CSCI 115. Chapter 5 Functions. CSCI 115. §5 .1 Functions. §5 .1 – Functions. Function Relation such that for every domain element a, |f(a)| = 1 Mappings, transformations f(a) = {b} f(a) = b a = argument b = function value. §5 .1 – Functions. Types of functions Everywhere defined

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CSCI 115

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  1. CSCI 115 Chapter 5 Functions

  2. CSCI 115 §5.1 Functions

  3. §5.1 – Functions • Function • Relation such that for every domain element a, |f(a)| = 1 • Mappings, transformations • f(a) = {b} • f(a) = b • a = argument • b = function value

  4. §5.1 – Functions • Types of functions • Everywhere defined • Onto • One to one (1–1) • 1 – 1 Correspondence • ED, Onto, 1–1 • Invertible Functions

  5. §5.1 – Functions • Theorem 5.1.1

  6. §5.1 – Functions • Theorem 5.1.2

  7. §5.1 – Functions • Theorem 5.1.4

  8. §5.1 – Functions • 1-1 functions and cryptography • Allows coding AND decoding • Substitution codes • Table from Example 18 (p. 188)

  9. CSCI 115 §5.2 Functions for Computer Science

  10. §5.2 – Functions for Computer Science • mod–n (or modn) • Factorial • Floor • Ceiling • Boolean • Hashing • Others

  11. §5.2 – Functions for Computer Science • Set • Collection of objects • Any element is unambiguously in the set or not • Characteristic function • Fuzzy sets • Whether or not an element is in the set may be ‘fuzzy’ • The set of all rich people

  12. §5.2 – Functions for Computer Science • Fuzzy sets • Function f defined on a set having values in the interval [0, 1] • If f(x) = 0, x is not in the set • If f(x) = 1, x is in the set • If 0 < f(x) < 1 then f(x) is the degree to which x is in the set • Degree of membership • Ordinary sets are special cases of fuzzy sets

  13. §5.2 – Functions for Computer Science • Fuzzy set operations • Theorem 5.2.1 (Finding other degrees of membership) • Let A and B be subsets of the same universal set U. Then:

  14. §5.2 – Functions for Computer Science • Fuzzy Logic • Fuzzy predicates • Values can be true, false, or somewhere in between • Schrodinger’s Cat • Quantum theory and computers • Applications • Control theory • Elevator operation • ABS systems in cars • Expert systems

  15. CSCI 115 §5.3 Growth of Functions

  16. §5.3 – Growth of Functions • Algorithmic Analysis • Efficiency • Number of steps (running time) • Comparison

  17. §5.3 – Growth of Functions • Definitions • Let f and g be functions whose domains are subsets of Z+. • We say f is O(g) (read f is big-Oh of g) if  constants c and k s.t. |f(n)|  c |g(n)|  n  k. • We say f and g have the same order if f is O(g) and g is O(f) • We say f is lower order than g (or f grows more slowly than g) if f is O(g) but g is not O(f)

  18. §5.3 – Growth of Functions • Definition • We define a relation Θ (called big-theta) on functions whose domains are subsets of Z+ as:fΘg iff f and g have the same order. • Theorem 5.3.1 • The relation Θ is an equivalence relation.

  19. §5.3 – Growth of Functions • Equivalence classes of Θ • Equivalence classes (called Θ–classes) consist of functions of the same order • One Θ–class is said to be lower than another if any of the functions in the first is lower than any in the second • Θ–classes provide the necessary information to do algorithmic analysis

  20. §5.3 – Growth of Functions • Image from page 203 of the text

  21. CSCI 115 §5.4 Permutation Functions

  22. §5.4 – Permutation Functions • Permutation • 1-1 correspondence from a set onto itself • Theorem 5.4.1 • If A = {a1, a2, …, an} with |A| = n, then n! permutations of A

  23. §5.4 – Permutation Functions • Cycle of length r • If p(a1) = a2, p(a2) = a3, …, p(ar-1) = ar and p(ar) = a1, this is called a cycle of length r, and is denoted (a1, a2,…, ar) • Disjoint cycles

  24. §5.4 – Permutation Functions • Theorem 5.4.2 • A permutation of a finite set that is not the identity or a cycle can be written as a product of disjoint cycles of length greater than or equal to 2

  25. §5.4 – Permutation Functions • Transposition • Cycle of length 2 • Every cycle can be written as a product of transpositions as follows: • (b1, b2, …, br) = (b1, br)(b1, br-1)  …  (b1, b2)

  26. §5.4 – Permutation Functions • Even permutation • A permutation that can be written as the product of an even number of transpositions • Odd permutation • A permutation that can be written as the product of an odd number of transpositions

  27. §5.4 – Permutation Functions • Theorem 5.4.3 • A permutation cannot be both even and odd • Theorem 5.4.4 • Let A = {a1, a2, …, an}, with |A| = n  2. Then there are n!/2 even permutations of A, and n!/2 odd permutations of A.

  28. §5.4 – Permutation Functions • Transposition codes • Encrypt by using the following transpositionMATH IS GREAT(1, 10, 7)  (11, 3, 2, 8)  (5, 4, 9) • Decrypt the followingMessage: OICLPCYIGULSOYRRVTTEATransposition Code: (16, 15, 6, 1, 3, 7, 11, 2)  (8, 13, 4, 5, 19)  (14, 10)  (21, 20, 17)

  29. §5.4 – Permutation Functions • Transposition codes • Keyword Columnar Transposition – Example 8Using the keyword “JONES” to encrypt:THE FIFTH GOBLET CONTAINS THE GOLDResult:FGTAHDTFBONGEHETTLHTLNSOIOCIEX

  30. §5.4 – Permutation Functions • Transposition codes • Keyword Columnar TranspositionUse the keyword “BASEBALL” to decrypt:AAK7ENWHRSOER9SAETOELNNWIDYELXBO1DXKTI3RUsing a keyword columnar transposition

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