CMSC 203 / 0201 Fall 2002

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CMSC 203 / 0201Fall 2002

• Week 12 11/13/15 November 2002
• Prof. Marie desJardins
• Guest lecturer/proctor Prof. Dennis Frey

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MON 11/11 RELATIONS (6.1-6.2)
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Concepts/Vocabulary

• Binary relation R ? A x B (also written a R b
  or R(a,b))
• Relations on a set R ? A x A
• Properties of relations
• Reflexivity (a, a) ? R
• Symmetry (a, b) ? R ?? (b, a) ? R
• Antisymmetry (a, b) ? R ? ab
• Transitivity (a, b) ? R ? (b, c) ? R ? (a, c) ?
  R
• Composite relation
• (a, c) ? S?R ? ?b?B (a, b)?R ? (b, c)? S
• Powers of a relation R1 R, Rn1 Rn ? R
• Inverse relation (b,a) ? R-1 ? (a,b) ? R
• Complementary relation (a,b) ?R ? (a,b) ? R

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Concepts/Vocabulary II

• n-ary relation R ? A1 x A2 x x An
• Ai are the domain of R n is its degree (or
  arity)
• In a database, the n-tuples in a relation are
  called records the entries in each record (i.e.,
  elements of the ith set in that n-tuple) are the
  fields
• In a database, a primary key is a domain (set Ai)
  whose value completely determines which n-tuple
  (record) is indicated i.e., there is only one
  n-tuple for a given value of that domain
• A composite key is the Cartesian product of a set
  of domains whose values completely determine
  which n-tuple is indicated
• Projection delete certain fields in every record
• Join merge (union) two relations using common
  fields

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Examples

• Exercise 6.1.4 Determine whether the relation R
  on the set of all people is reflexive, symmetric,
  antisymmetric, and/or transitive, where (a,b) ? R
  iff
• (a) a is taller than b
• (b) a and b were born on the same day
• (c) a has the same first name as b
• (d) a and b have a common grandparent

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Examples II

• Exercise 6.1.21 Let R be the relation on the set
  of people consisting of pairs (a, b) where a is a
  parent of b. Let S be the relation on the set of
  people consisting of pairs (a, b) where a and b
  are siblings (brothers or sisters). What are S ?
  R and R ? S?
• Exercise 5.1.29 Show that the relation R on a
  set A is symmetric iff R R-1.
• Exercise 5.1.33 Let R be a relation that is
  reflexive and transitive. Prove that Rn R for
  all positive integers n.

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WED 11/13RELATIONS II (6.3-6.4)
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Concepts / Vocabulary

• Zero-one matrix representation of binary
  relations
• Matrix interpretations of properties of relations
  on a set reflexivity, symmetry, antisymmetry,
  and transitivity
• Digraph representation of binary relations
• Pictorial interpretations of properties of
  relations on a set
• Closure of R with respect to property P
• smallest relation containing R that satisfies P
• Transitive closure, reflexive closure,
• Path analogy for transitive closures
  connectivity relation R Algorithm 6.4.1 for
  computing transitive closure (briefly)
  Warshalls algorithm (briefly)

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Examples

• Exercise 6.3.5 How can the matrix for R be found
  from the matrix representing R, a relation on a
  finite set A?
• Exercise 6.3.15 List the ordered pairs in the
  relations represented by the directed graphs

b
a
d
c
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• Exercise 6.3.18 (partial) Given the digraphs
  representing two relations, how can the directed
  graphs of the union and difference of these
  relations be found?
• Exercise 6.4.8 How can the directed graph
  representing the symmetric closure of a relation
  on a finite set be constructed from the directed
  graph for this relation?
• Exercise 6.4.23 Suppose that the relation R is
  symmetric. Show that R is symmetric.

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FRI 11/8MIDTERM 2

• Good luck! ?