Chapter 2
51
10.
This follows quickly from one
of
the laws
of
De Morgan and the identity X "
Y
=
X
n
YC.
A"
(B
n
C)
=
An
(B
n
C)C
=
An
(B
C
U
CC)
::l::
(A
nBC)
U
(A
n
CC)
=
(A"
B)
U
(A
"
C),
using (3), p. 62 at the spot marked with the arrow.
11.
(a) A
binary relation
on
A
is a subset
of
Ax
A.
(b)
If
A
has 10 elements,
A
x
A
has 100 elements, so there are 2
100
binary relations on
A.
12.
Reflexive: For any
a
with
lal
~
1, we have
a
2
=
la
2
1
=
lallal
~
lal,
thus
(a, a)
E
R.
Symmetric by definition.
Not antisymmetric because
(1 1)
is in
R
«(1)2
<
1
and
(1)2
<
1)
but
1
'
1
2'
4
2

4
4

2
2
.,
4·
Not transitive. We have
(~,
PER
because
(~)2
~
1
and
(1)2
~
~
and
(1, g)
E
R
because
(1)2
~
g
and
(g)2
~
1,
but
(2'
g)
is not in
R
because
(~)21,
g.
13.
(a) Reflexive: For any natural number
a,
we have
a
~
2a,
so
a
'"
a.
Not symmetric: 2
'"
5 because 2
~
2(5), but 5
rf
2 because 5
1,
2(2).
Not antisymmetric: Let
a
=
1 and
b
=
2.
Then
a
'"
b
because 1
~
2(2) and also
b
'"
a
because
2
~
2(1).
Not transitive: Let
a
=
3,
b
=
2, c
=
1.
Then
a
'"
b
because 3
~
2(2) and
b
'"
c because
2
~
2(1). However,
a
rf
c because
31,
2(1).
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 Summer '10
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 Graph Theory, Equivalence relation, Binary relation, Transitive relation, Symmetric relation, Preorder

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