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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
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory

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