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CMSC 203 / 0201 Fall 2002PowerPoint Presentation

CMSC 203 / 0201 Fall 2002

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### CMSC 203 / 0201Fall 2002

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

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

### FRI 11/8 two relations, how can the directed graphs of the union and difference of these relations be found?MIDTERM #2

Week #12 – 11/13/15 November 2002

Prof. Marie desJardins

Guest lecturer/proctor: Prof. Dennis Frey

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 a=b
- Transitivity: (a, b) R (b, c) R (a, c) R

- Composite relation:
- (a, c) SR bB: (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

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

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

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.

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); Warshall’s algorithm (briefly)

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

- 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!

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