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
This is the end of the preview. Sign up
to
access the rest of the document.
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, Sets

Click to edit the document details