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COT 3100 Homework #4 Fall ’00
Lecturer: Arup Guha
Assigned: 10/17/00
Due: 10/24/00 (No late assignments)
1)
Prove or disprove: If a relation R, defined as a subset of A x A, is transitive, then R
°
R
is transitive as well.
2)
Define sumdigits(a), where a is a positive integer, to be the sum of the digits in the
decimal representation of the positive integer a. Using this definition of the function
sumdigits, consider the following relation:
{(a,b)  a
∈
Z
+
∧
b
∈
Z
+
∧
(a
≥
b
∨
sumdigits(a)
≥
sumdigits(b)}
Determine if this relation is (i)reflexive, (ii)irreflexive, (iii)symmetric, (iv)antisymmetric,
and (v)transitive?
The rest of the questions will be based on the following definitions. (Note: Prove
your answer to questions 3, 4, or 5 through a valid proof or a disproof using a
counterexample.)
Definition.
Let
R
⊂
A
×
A
denote a binary relation.
The following relations defined
over
A
are called
closures
:
(a)
The
reflexive closure
of
R
is
r
(
R
) =
R
∪
{(
a
,
a
) 
a
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 Fall '09

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