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# CMSC 203 / 0201 Fall 2002 - PowerPoint PPT Presentation

CMSC 203 / 0201 Fall 2002. Week #12 – 11/13/15 November 2002 Prof. Marie desJardins Guest lecturer/proctor: Prof. Dennis Frey. MON 11/11 RELATIONS (6.1-6.2). 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

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### CMSC 203 / 0201Fall 2002

Week #12 – 11/13/15 November 2002

Prof. Marie desJardins

Guest lecturer/proctor: Prof. Dennis Frey

### MON 11/11 RELATIONS (6.1-6.2)

• 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  a=b

• 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, Rn+1 = Rn  R

• Inverse relation: (b,a)  R-1  (a,b)  R

• Complementary relation: (a,b) R  (a,b)  R

• 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

• 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

• 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.

### WED 11/13RELATIONS II (6.3-6.4)

• 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); Warshall’s algorithm (briefly)

• 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

• 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.

Good luck! 