Li 1
Student Name: Kaitian Li
Professor: Daniel Vogel
History 1301-23417
April 24, 2017
Dances with Wolves
Dances with Wolves is a American historical Western film, and it was directed,
produced and starring by Kevin Costner, The film won seven Academy Aw
Name: Kaitian Li
Professor: Daniel Vogel
HIST 1301 23417
April 20, 2017
Threats to Liberty
The document from President George Washington was from that he announced he
would not be a candidate for reelection in 1796, also we call George Washingtons
farewel
an infinite set. Solution: The pair (1, 1)
is in R1, R3, R4, and R6; (1, 2) is in R1
and R6; (2, 1) is in R2, R5, and R6; (1,
1) is in R2, R3, and R6; and finally, (2,
2) is in R1, R3, and R4. It is not hard
to determine the number of relations
on a finit
computer programs. On the other
hand, people often find the
representation of relations using
directed graphs useful for
understanding the properties of these
relations. Representing Relations
Using Matrices A relation between
finite sets can be represent
resulting subgraph has the same vertex
set V as G. Its edge set is E E . P0 P1
P2 P3 P4 P5 P6 P7 FIGURE 14 A
Hypercube Network for Eight
Processors. d c a e b c a b e FIGURE 15
A Subgraph of K5. P1: 1 CH10-7T
Rosen-2311T MHIA017-Rosen-v5.cls
May 13, 2011
use mathematical induction to prove
the only if part of the theorem. Note
that this part of the theorem is trivially
true for n = 1. Assume that Rn R,
where n is a positive integer. This is the
inductive hypothesis. To complete the
inductive step we must
Which projection mapping is used to
delete the first, second, and fourth
components of a 6-tuple? 16. Display
the table produced by applying the
projection P1,2,4 to Table 8. 17.
Display the table produced by applying
the projection P1,4 to Table 8. 18. H
Rosen-2311T MHIA017-Rosen-v5.cls
May 13, 2011 15:29 9.1 Relations and
Their Properties 579 pairs of the form
(a, b), where a = b, may or may not be
in R. Hence, by the product rule for
counting, there are 2n (n1) reflexive
relations [this is the number of
viable order for the different phases of
a complicated project, or producing a
useful way to store information in
computer databases. In some
computer languages, only the first 31
characters of the name of a variable
matter. The relation consisting of
ord
on n-ary relations. EXAMPLE 12 We
will illustrate how SQL is used to
express queries by showing how SQL
can be employed to make a query
about airline flights using Table 8. The
SQL statement SELECT Departure_time
FROM Flights WHERE
Destination=Detroit is
Recall that a function f from a set A to a
set B (as defined in Section 2.3) assigns
exactly one element of B to each
element of A. The graph of f is the set of
ordered pairs (a, b) such that b = f (a).
Because the graph of f is a subset of A
B, it is a
every person x. DEFINITION 3 A
relation R on a set A is called reflexive
if (a, a) R for every element a A.
Remark: Using quantifiers we see that
the relation R on the set A is reflexive if
a(a, a) R), where the universe of
discourse is the set of all ele
section. More information can be
found in [AhUl95]. Exercises 1. List the
triples in the relation cfw_(a, b, c) | a, b,
and c are integers with 0 <a<b<c< 5. 2.
Which 4-tuples are in the relation cfw_(a,
b, c, d) | a, b, c, and d are positive
integers with
Marcus Mathematics Miller Computer
Science EXAMPLE 8 What results when
the projection P1,3 is applied to the 4tuples (2, 3, 0, 4), (Jane Doe,
234111001, Geography, 3.14), and (a1,
a2, a3, a4)? Solution:The projection
P1,3 sends these 4-tuples to (2, 0),
(
binary relation from A to B is a set R of
ordered pairs where the first element
of each ordered pair comes from A and
the second element comes from B. We
use the notation aRb to denote that (a,
b) R and a R b to denote that (a,
b) / R. Moreover, when (a,
relationships between sets. (Recall
that the graph of the function f from A
to B is the set of ordered pairs (a, f (a)
for a A.) P1: 1 CH09-7T Rosen-2311T
MHIA017-Rosen-v5.cls May 13, 2011
15:29 9.1 Relations and Their
Properties 575 Relations on a Set
Re
every integer a), R3, and R4. For each
of the other relations in this example it
is easy to find a pair of the form (a, a)
that is not in the relation. (This is left
as an exercise for the reader.)
EXAMPLE 9 Is the divides relation on
the set of positive
publication date) be a composite key?
c) Under what conditions would (title,
number of pages) be a composite key?
P1: 1 CH09-7T Rosen-2311T MHIA017Rosen-v5.cls May 13, 2011 15:29 590
9 / Relations 9. The 5-tuples in a 5-ary
relation represent these attrib
on the set A = cfw_a1, a2,.,an is
symmetric if and only if (aj , ai) R
whenever (ai, aj ) R. In terms of the
entries of MR, R is symmetric if and
only if mj i = 1 whenever mij = 1. This
also means mj i = 0 whenever mij = 0.
Consequently, R is symmetric if
(Ackermann, 231455, Computer
Science, 3.88) and (Rao, 678543,
Mathematics, 3.90). Finally, to find the
records of computer science majors
who have a GPA above 3.5, we use the
operator sC3 , where C3 is the
condition (Major = Computer Science
GPA > 3.5).
of a composite of two relations.
DEFINITION 7 Let R be a relation on
the set A. The powers Rn, n = 1, 2, 3,.,
are defined recursively by R1 = R and
Rn+1 = Rn R. The definition shows
that R2 = R R, R3 = R2 R = (R R) R,
and so on. EXAMPLE 22 Let R = cfw_(1,
EXAMPLE 18 Let A and B be the set of
all students and the set of all courses at
a school, respectively. Suppose that R1
consists of all ordered pairs (a, b),
where a is a student who has taken
course b, and R2 consists of all ordered
pairs(a, b), where a
u or v and with the same second
endpoint. Hence, the contraction of the
edge e with endpoints u and v in the
graph G = (V , E) produces a new graph
G = (V , E ) (which is not a subgraph of
G), where V = V cfw_u, vcfw_w and
Econtains the edges in E which d
have shown is that R has the transitive
property, which is defined as follows.
DEFINITION 5 A relation R on a set A is
called transitive if whenever (a, b) R
and (b, c) R, then (a, c) R, for all a,
b, c A. Remark: Using quantifiers we
see that the relatio
Pi1,i2,.,im (S). 26. Give an example to
show that if R and S are both n-ary
relations, then Pi1,i2,.,im (R S) may
be different from Pi1,i2,.,im (R)
Pi1,i2,.,im (S). 27. Give an example to
show that if R and S are both n-ary
relations, then Pi1,i2,.,im (R
transitive because a>b and b>c imply
that a>c. R3 is transitive because a = b
and b = c imply that a = c. R4 is
clearly transitive, as the reader should
verify. R5 is not transitive because (2,
1) and (1, 0) belong to R5, but (2, 0)
does not. R6 is not tr
the set of all integers is reflexive,
symmetric, antisymmetric, and/or
transitive, where (x, y) R if and only
if a) x = y. b) xy 1. c) x = y + 1 or x = y
1. d) x y (mod 7). e) x is a multiple of
y. f ) x and y are both negative or both
nonnegative. g) x
What do isolated and pendant vertices
represent? 14. What does the degree of
a vertex in the Hollywood graph
represent? What does the
neighborhood of a vertex represent?
What do the isolated and pendant
vertices represent? 15. What do the indegree and the