Mathematics for Computer Science

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Mathematics for Computer Science

• Lecture 4 Sets and Numbers
• Leon van der Torre

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Slides

• Slides can be found on the internet
• These slides are based on MIT open courseware,
  by S. Devadas and E. Lehmann

BECS - Bachelor of Engineering in Computer
Science
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Part 1 Set theory

• A set is a collection of distinct objects called
  elements.
• Lists the elements inside curly-braces.
• N  0,1,2,3,... the natural numbers
• C  red,blue,yellow primary colors
• D  Nifty,Friend,Horatio,Pretty-Pretty dead
  pets
• P  a,b ,a,c ,b,c a set of sets
• Elements of a set are required to be distinct
• Order of elements is not significant, so x,y
  y,x
• Expression e ? S asserts that e is an element of
  set S.
• 7 ? N and blue ? C, but Wilbur ? D yet.
• Sets are simple, flexible, and everywhere.

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Some popular sets

• Special symbols to represent some common sets.
• A superscript  restricts a set to its positive
  elements
• for example, R is the set of positive real
  numbers.

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Comparing sets

• S ? T indicates that set S is a subset of set T,
• N ? Z and Q ? R, but not C ? Z
• S ? T S is a subset of T, but they are not
  equal.
• So for every set A, A ? A, but not A ? A.

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Combining sets. X  1,2,3,Y  2,3,4

• The union of sets X and Y  is denoted X ? Y
• all elements appearing in X or Y or both X ?
  Y  1,2,3,4.
• The intersection of X and Y  is denoted X ? Y
• all elements that appear in both X and Y X ?
  Y  2,3.
• The difference of X and Y  is denoted X - Y
• all elements in X, but not in Y X - Y  1 ,
  Y - X  4.
• The complement of a set Z,
• consists of all elements not in Z, given a larger
  set.
• with the real numbers R, the complement of the
  positive real numbers is the set of negative real
  numbers together with zero

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Sequences

• A sequence is a list of objects called terms
• Listing the elements between parentheses
• (a,b,c) is a sequence with three terms.
• Terms in a sequence can be the same.
• (a,b,a) is valid sequence, but a,b,a is not
  valid set
• The terms in a sequence have a specified order.
• (a,b,c) and (a,c,b) are different sequences, but
  a,b,c a,c,b
• The empty sequence is typically ?.

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Links between sets and sentences

• A product of sets, S1  S2  ...  Sn, is a new
  set consisting of all sequences where the first
  term is drawn from S1, the second from S2, and so
  forth.
• N N  (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...
  
• A x B A x B B x A
• Product of n copies of set S is denoted Sn.
• 0,13  
• (0,0,0),(0,0,1),(0,1,0),(0,1,1),(1,0,0),(1,0,1),(
  1,1,0),(1,1,1)

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Set Builder Notation

• to define a set using a predicate
• the set consists of all values that make the
  predicate true
• A  n ? N n is prime and n 4k 1 for integer
  k
• the elements are 5, 13, 17, 29, 37, 41, 53, 57,
  61, 73, ...
• B   x ? R x3 - 3x 1 gt 0 
• C   a  bi  ? C a2  2b2  1
• Oval-shaped region around the origin in the
  complex plane

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Characteristic function

• fA 1 if x ?A
• 0 otherwise
• fA ? B fA x fB
• fA ? B fA fB - fA x fB
• fnot A 1 - fA
• fA-B fA x (1 - fB)

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Infinite sets

• Infinite set there exists a 1-1 mapping on a
  proper subset
• Countable there is a 1-1 mapping on N
• Q is countable

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Properties of set operations

• P and Q are sets
• (P ? Q) (Q ? P)
• P ? (Q ? R) (P ? Q) ? R
• P ? (Q ? R) (P ? Q) ? (P ? R)
• (P ? P) P
• not not P P
• not (P ? Q) not P ? not Q

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This is a Boolean algebra

• P and Q are propositions
• (P ? Q) ? (Q ? P)
• P ? (Q ? R) ? (P ? Q) ? R
• P ? (Q ? R) ? (P ? Q) ? (P ? R)
• (P ? P) ? P
• ??P ? P
• ? (P ? Q) ? ?P ? ?Q

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Part 2 Number theory

• P and Q are integers
• (P x Q) (Q x P)
• P x (Q x R) (P x Q) x R
• P x (Q R) (P x Q) (P x R)
• But not, e.g.
• P (Q x R) (P Q) x (P R)
• Are there any problems with integers?

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Divisibility

• a divides b if there is an integer k such that a
  x k  b.
• This is denoted a  b.
• For example7 63 because 7x9 63 
• Every number divides zero since a x 0  0 for
  every integer a.
• If a divides b, then b is a multiple of a.
• For example, 63 is a multiple of 7.

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Facts About Divisibility

• If a  b, then a  bc for all c.
• If a  b and b c, then a c.
• If a  b and a c, then a  sb tc for all s  t.
• For all c ? 0, a  b if and only if ca  cb.

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When Divisibility Goes Bad

• More precisely, if you divide n by d, then you
  get a quotient q and a remainder r.
• Theorem (Division Theorem). Let n and d be
  integers such that d gt 0. Then there exists a
  unique pair of integers q and r such that n 
  qd r and 0 r lt d.
• d  10, n  2716 q  271 and r 6,
• since 2716  271  10  6.
• We write r n mod d

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The Greatest Common Divisor

• The largest number that is a divisor of both
  a and b. It is denoted gcd(a, b).
• For example, gcd(18, 24)6.
• Theorem. The greatest common divisor of a and
  b is equal to the smallest positive linear
  combination of a and b.
• gcd(52,44) 4.
• 4 is a linear combin. of 52 and 446 x
  52  (-7) x 44  4
• Furthermore, no linear combination of 52 and 44
  is equal to a smaller positive integer.

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Properties Greatest Common Divisor

• Every common divisor of a and b divides
  gcd(a, b).
• gcd(ka, kb) k  gcd(a, b) for all kgt 0.
• If gcd(a, b) 1 and gcd(a, c)1, then
  gcd(a, bc) 1.
• If a  bc and gcd(a, b) 1, then ac.
• gcd(a, b)  gcd(b, a mod b).
• Can be used to compute the gcd efficiently

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Fundamental Theorem of Arithmetic

• Every positive integer n can be written in a
  unique way as a product of primes
• n  p1 p2 pj (p1  p2  ...  pj)
• 1 is not a prime 15  3x5 or 1x3x5 or 12 35
• the product of an empty set of numbers is defined
  to be 1 (for n1)
• Lemma. If p is a prime and p ab, then p a or
  p b.

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Application of number theory

• Cryptography

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Homework

• Outils mathématiques pour l'informaticien, pages
  31-75
• Read section 2.1 - 2.5
• Do exercises 2.1 - 2.4