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Section 3.3
67
Antisymmetric:
If
A, B
E
P(S), A
~
Band B
~
A,
then we have
IAI
~
IBI
and
IBI
~
IAI,
so
IAI
=
IBI.
Since
P(S)
does not contain different sets of the same cardinality, it follows that
A
=
B.
Transitive: Suppose
A
~
Band B
~
C.
If
A
=
0,then
IAI
=
°
~
101
no matter what
C
is, so we'd
have
A
~
C.
If
A
=
S,
then
A
~
B
means
B
=
Sand B
~
C
means
C
=
S,
so
A
=
B
=
C
=
S
and
A
~
C.
9.
I:
A
x
B
+
B
x
A
defined by
I(a,
b)
=
(b,
a)
is a onetoone onto function.
10. (a) [BB] False. Let X
=
{I},
Y
=
{2},
Z
=
{3}. Then ((1,2),3) E (X x
Y)
x
Z
but ((1,2),3)
fj.
X x
(Y
x
Z)
(a set whose second coordinates are ordered pairs).
(b) [BB] Define
I:
(X x
Y)
x
Z
+
X x
(Y
x
Z)
by
I((x, y), z)
=
(x, (y, z)).
11. (a) Reflexivity: For any set
A,
"A
is a onetoone onto function
A
+
A,
so
A
has the same cardinality
as itself.
Symmetry:
If
A
and
B
have the same cardinality, then there is a onetoone onto function
I:
A
+
B.
Such a function has an inverse
11
:
B
+
A
which is onetoone and onto (because it has an
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 Summer '10
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 Graph Theory, Sets

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