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CSCI 115

CSCI 115. Chapter 1 Fundamentals. CSCI 115. § 1.1 Sets and Subsets. § 1.1 – Sets and Subsets. Definition Element Order does not matter Special Sets (Z, Z + , Z – , Q , , R , C , {}) Other Sets Equality Subsets Cardinality Power Set. CSCI 115. § 1.2 Operations on Sets.

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CSCI 115

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  1. CSCI 115 Chapter 1 Fundamentals

  2. CSCI 115 §1.1 Sets and Subsets

  3. §1.1 – Sets and Subsets • Definition • Element • Order does not matter • Special Sets (Z, Z+, Z–, Q, , R, C, {}) • Other Sets • Equality • Subsets • Cardinality • Power Set

  4. CSCI 115 §1.2 Operations on Sets

  5. §1.2 – Operations on Sets • Union • Intersection • Special notation • Disjoint Sets • Complement (B with respect to A) • Symmetric Difference

  6. §1.2 – Operations on Sets • Theorem 1.2.2 • Addition Principle for 2 sets: |A  B| = |A| + |B| - |A  B| • Theorem 1.2.3 • Addition Principle for 3 sets: |A  B  C| = |A| + |B| + |C| - |A  B| - |B  C| - |A  C| + |A  B  C|

  7. CSCI 115 §1.3 Sequences

  8. §1.3 – Sequences • List of objects in a definite order • Finite • Infinite • Recursively defined • Explicitly defined • Arrays

  9. §1.3 – Sequences • Characteristic Function • For a set A, fA(x) = 1 xA 0 xA • Theorem 1.3.1 • Characteristic functions satisfy the following • i) fAB = fAfB • ii) fAB = fA + fB –fAfB • iii) fA⊕B = fA + fB – 2fAfB {

  10. §1.3 – Sequences • Computer representations of sets • Utilizes the characteristic function • Universal set must be clearly defined • Stored in Arrays • U = {1, 2, 3, 4, 5, 6}A = {1, 3, 6} • Computer representation of A • fA = {1, 0, 1, 0, 0, 1}

  11. §1.3 – Sequences • Cardinality • Finite • Infinitely countable • Infinitely uncountable

  12. §1.3 – Sequences • Strings and Regular Expressions • Given a set A, A* is the set of all finite sequences of elements in A (  A*) • A – alphabet • A* – set of words • Concatenation

  13. §1.3 – Sequences • Regular Expressions – a regular expression over A is a string constructed from the elements of A, (, ), v, *, and  according to: • RE1:  is a regular expression • RE2: If x  A, x is a regular expression • RE3: If a and b are regular expressions, then ab is a regular expression • RE4: If a and b are regular expressions, then a v b is a regular expression • RE5: If a is a regular expression, then a* is a regular expression • We will not be covering regular subsets

  14. CSCI 115 §1.4 Properties of Integers

  15. §1.4 – Properties of Integers • Theorem 1.4.1 • If n and m are integers and n > 0, we can write m = qn + r for integers q and r with 0  r < n. Moreover, there is just one way to do this. • If the r in theorem 1.4.1 is 0, we say n divides m, and write n|m. Then m = qn with n m.

  16. §1.4 – Properties of Integers • Theorem 1.4.2 • Let a, b, and c be integers • If a|b and a|c, then a|(b + c) • If a|b and a|c, where b > c, then a|(b – c) • If a|b or a|c, then a |bc • If a|b and b|c, then a|c

  17. §1.4 – Properties of Integers • A integer p is prime if p > 1 and the only positive integers that divide p are p and 1 • Theorem 1.4.3 • Every positive integer n > 1 can be written uniquely as , where p1<p2<…<ps are distinct primes that divide n and the k’s are positive integers giving the number of times each prime occurs as a factor of n

  18. §1.4 – Properties of Integers • Greatest Common Divisor • If a, b, and k are in Z+, and k|a and k|b, we say that k is a common divisor of a and b. If d is the largest such k, d is called the greatest common divisor, or GCD, of a and b, and we write d = GCD(a, b).

  19. §1.4 – Properties of Integers • Theorem 1.4.4 • If d is GCD(a, b), then • d = sa + tb for some integers s and t. (s and t may not be positive) • If c is any other common divisor of a and b, then c|d • Theorem 1.4.5 • If a and b are in Z+, then: GCD(a, b) = GCD(b, ba)

  20. §1.4 – Properties of Integers • Algorithm 1 to find GCD(a, b) (assume a > b)Euclidean Algorithm • Find q and r such that a = qb + r (as in Thm 1.4.1) • Replace a with b, b with r • Continue process from step 1 until r = 0 • The GCD is the last of the non-zero divisors Ex: Find GCD (5797, 68355) using Algorithm 1

  21. §1.4 – Properties of Integers • Algorithm 2 to find GCD(a, b) • Find the prime factorizations of a and b • Find the product of only those prime numbers represented in both factorizations, to the lowest power represented Ex: Find GCD (5797, 68355) using Algorithm 2

  22. §1.4 – Properties of Integers • Least Common Multiple • If a, b, and k are in Z+, and a|k and b|k, we say k is a common multiple of a and b. The smallest such k, called c, is called the least common multiple, or LCM, of a and b, and we write c = LCM(a, b).

  23. §1.4 – Properties of Integers • Theorem 1.4.6 • If a and b are two positive integers, then GCD(a, b) · LCM(a, b) = ab

  24. §1.4 – Properties of Integers • Algorithm to find LCM(a, b) • Find the prime factorizations of a and b • Find the product each of the prime numbers represented in either factorization, to the greatest power represented Ex: Find LCM (5797, 68355)

  25. §1.4 – Properties of Integers • Theorem 1.4.7 • If b > 1 is an integer, then every positive integer n can be uniquely expressed in the form:where This is called the base b expansion of n. • Example: Find the base 7 representation of 381.

  26. §1.4 – Properties of Integers • Cryptology – Sir Francis Bacon’s codeDISCRETE00011 01000 10010 00010 10001 00100 10011 00100OREGO NISAS TATEI NTHEU NITED STATE SOFAM ERICAMessage sent:OREGON ISASTATE INTHE UNITED STATES OF AMERICA

  27. §1.4 – Properties of Integers • Example – decode the followingWILLIAMIS A FAMOUSAUTHOR FROMTHE SIXTEENTHCENTURYKey: Underlined and bold text stands for 0 • Steganography

  28. CSCI 115 §1.5 Matrices

  29. §1.5 – Matrices • Matrix – Rectangular array • ith row, jth column, i,j element • Square matrix, diagonal • Diagonal matrix • Equality • Zero Matrix (additive identity) • Identity Matrix (multiplicative identity)

  30. §1.5 – Matrices • Addition • Theorem 1.5.1 • i) A + B = B + A • ii) (A + B) + C = A + (B + C) • iii) A + 0 = 0 + A = A

  31. §1.5 – Matrices • Multiplication • Theorem 1.5.2 • i) A(BC) = (AB)C • ii) A(B + C) = AB + AC • iii) (A + B)C = AC + BC

  32. §1.5 – Matrices • Commutativity of Multiplication? • Let A be size m x p, B be size p x n • BA: • May not be defined • May be defined, but a different size than AB • May be defined, same size as AB, but ABBA • May be equal to AB

  33. §1.5 – Matrices • Other properties / definitions: • If A is m x n, then ImA = AIn = A • If A is square (n x n): • Ap = AAA…A (p factors) • A0 = In • ApAq = A(p+q) • (Ap)q = Apq • (AB)p = ApBp if and only if AB = BA

  34. §1.5 – Matrices • Transposition • Theorem 1.5.3 • i) (AT)T = A • ii) (A + B)T = AT + BT • iii) (AB)T = BTAT • Symmetry (AT = A) • A is symmetric if and only if ai,j = aj,i for all i and j

  35. §1.5 – Matrices • Boolean Matrices (all elements are 0 or 1) • Operations on Boolean Matrices: • Let A and B be boolean Matrices • The join of A and B (C = A  B): • Ci,j = 1 if Ai,j = 1 or Bi,j = 1 • Ci,j = 0 if Ai,j = 0 and Bi,j = 0 • The meet of A and B (C = A  B): • Ci,j = 1 if Ai,j = 1 and Bi,j = 1 • Ci,j = 0 if Ai,j = 0 or Bi,j = 0

  36. §1.5 – Matrices • Boolean Matrices (all elements are 0 or 1) • Operations on Boolean Matrices: • Let A and B be boolean Matrices • The boolean product of A (m x p) and B (p xn) is (C = A  B): • Ci,j = 1 if Ai,j =1 and Bk,j = 1 for some k, 1  k  p • Ci,j = 0 otherwise

  37. §1.5 – Matrices • Theorem 1.5.4 (Inverses – not discussed) • Theorem 1.5.5If A, B, and C are boolean matrices of appropriate sizes, then:i) A  B = B  A A  B = B  Aii) (A  B)  C = A  (B  C) (A  B)  C = A  (B  C)iii) A  (B  C) = (A  B)  (A  C) A  (B  C) = (A  B)  (A  C) • iv) (A ⊙ B) ⊙ C = A ⊙ (B ⊙ C)

  38. CSCI 115 §1.6 Mathematical Structures

  39. §1.6 – Mathematical Structures • Mathematical structure (system) • A collection of objects with an operation or operations defined on those objects

  40. §1.6 – Mathematical Structures • Types of operations • Unary – operates on a single object • Binary – operates on two objects • Properties of operations • Closure • Commutativity • Associativity • Distribution of one over another • De Morgan’s laws for a unary operation * and binary operations  and  • (x  y)* = x*  y* and (x  y)* = x *  y*

  41. §1.6 – Mathematical Structures • Identities • A structure with binary operation  may contain a distinguished object e, with the property x  e = e  x = x for all x in the collection. We call e an identity for . • Theorem 1.6.1 • If e is an identity for a binary operation , then e is unique.

  42. §1.6 – Mathematical Structures • Inverses • If a binary operation has an identity e, we say y is a -inverse of x if x  y = y  x = e. • Theorem 1.6.2 • If  is an associative operation and x has a -inverse y, then y is unique.

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