Section 2.3
37
7.
[BB]
The argument assumes that for
a
E
n
there exists a
b
such that
(a,
b)
En.
This need not be the
case: See Exercise 5(g).
8.
(a)
[BB]
Reflexive: Every word has at least one letter in common with itself.
Symmetric:
If
a
and
b
have at least one letter in common, then so
do
b
and
a.
Not
antisymmetric:
(cat, dot) and (dot, cat)
are
both in the relation but
dot
=F
cat!!
Not
transitive: (cat, dot) and (dot, mouse)
are
both in the relation but (cat, mouse) is not.
(b) Reflexive: Let
a
be a person.
If
a
is not enrolled at Miskatonic University, then
(a,
a)
E
n.
On
the other hand,
if
a
is enrolled at MU, then
a
is taking at least one course with himself, so again
(a,
a)
En.
Symmetric:
If
(a,
b)
E
n,
then either
it
is the case that neither
a
nor
b
is enrolled at
MU
(so
neither is
b
or
a,
hence,
(b,
a)
E
n)
or
it
is the case that
a
and
b
are both enrolled and
are
taking
at least one course together
(in
which
caseb
and
a
are
enrolled and taking a common course, so
(b,
a)
En).
In any case,
if
(a,
b)
En,
then
(b,
a)
En.
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 Summer '10
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 Graph Theory, Transitive relation, nd, Miskatonic University

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