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COT3100.01, Fall 2000
S. Lang (11/02/2000)
Solution Key to Test #2
Nov. 11, 2000
Part I. (36 pts.) True/False questions.
(No explanation is needed.)
1.
Let
A
= {1, 2, 3, 4} and
R
⊂
A
×
A
denote a binary relation depicted by the directed graph given in
the figure.
Answer each of the following True/False questions:
2.
Let
R
⊂
A
×
A
and
T
⊂
A
×
B
denote two arbitrary binary relations.
Answer each of the following
True/false questions:
(a)
If
R
is transitive, then
R

1
is transitive.
True
, because if (
a
,
b
), (
b
,
c
)
∈
R

1
, then (
b
,
a
), (
c
,
b
)
∈
R
,
thus (
c
,
a
)
∈
R
because
R
is transitive by assumption.
Therefore, (
a
,
c
)
∈
R

1
.
(b) The reflexive closure
r
(
R
) is irreflexive.
False
, because (
a
,
a
)
∈
r
(
R
) for all
a
∈
A
.
(c)
(
R
ο
T
)
⊂
(
R
ο
R
ο
T
).
False
, for example,
R
= {(1, 2)},
T
= {(2, 3)}.
Then
R
ο
R
=
∅
,
R
ο
T
= {(1,
3)}, and
R
ο
R
ο
T
=
∅
.
(d) If
R
⊂
R

1
, then
R
is symmetric.
True
, if (
a
,
b
)
∈
R
, then (
a
,
b
)
∈
R

1
because
R
⊂
R

1
by
assumption.
Thus, (
b
,
a
)
∈
R
.
3.
Let
f
:
A
→
B
be a surjection and let
g
:
B
→
C
be a surjection, where
A
,
B
,
C
denote finite sets.
Answer the following True/false questions:
(a)
The composed function
g
ο
f
:
A
→
C
is a surjection.
True
, this is a theorem.
(b) 
A

≥

C
.
True
, because 
A

≥

B

≥

C
, by the counting principle.
4.
Let
R
⊂
A
×
A
denote a binary relation. Answer the following True/false questions:
(a)
If
R
defines a function, and the inverse relation
R

1
also defines a function, then
R
is an injection.
True
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 Fall '09

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