Part Ill: Relations (50 pt.) (30 pt, 5 pt. each) For each of the following relat
ID: 3603689 • Letter: P
Question
Part Ill: Relations (50 pt.) (30 pt, 5 pt. each) For each of the following relations, determine whether the relation is . A partial order .A striet order . Reflexive Ant-reneriv . Symmetric . Anti-symmetric A total order. . An equivalence relation Transitive a. RIs a relation on the set (1. 2. 3. 4} such that R-((1, 1), (1.2), 0.1). (2.2),(3,3), (4,4)). b. R is a relation on the set of all people such that (a. b) R If and only if a and b have a c. R is a relation on the power set of a set A such that (x. y) R if and only if xcy d. R is a relation on Z' such that (x. y) E R if and only if x divides y e. Risarelation on z such that (x,y) E R if and only if r # y. t. R is a relation on Z such that (x, y) E R If and only if xy 21.Explanation / Answer
A “relation” is a mathematical tool used to describe relationships between set elements.
Relations are widely used in computer science, especially in databases and and schedul-
ing applications.
We’re mainly going to focus on relationships between elements of a single set; that is,
relations from a set A to a set B where A=B. Thus, a relation on a set A is a subset RA*A.
Let
A
be a set of people and the relation
R
describe who likes whom; that is,
(
x, y
)
R
if and only if
x
likes
y
.
•
Let
A
be cities. Then we can define a relation
R
such that
xRy
if and only if there is
a nonstop flight from city
x
to city
y
.
•
Let
A
=
Z
, and let
xRy
hold if and only if
x
y
(mod 5)
.
•
Let
A
=
N
, and let
xRy
hold if and only if
x
|
y
.
•
Let
A
=
N
, and let
xRy
hold if and only if
x
y