CS 336 PreTest 3 Solutions
Relations and Functions
4. Relations
1)
* For each of the following sets, state whether or not it is a partition of {0, 1, 2, 3, 4, 5, 6, 7,
8, 9, 10}.
a)
{{0}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {10}}
This is a partition
b)
{
∅
, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {10}}
This is a not partition since it contains the empty set and also has no subset containing 0.
c)
{{1, 2}, {3, 4}, {5, 6}, {7, 8}, {9, 10}}
This is a not partition since it has no subset containing 0.
d)
{{1, 2}, {2, 3}, {3, 4}, {4, 5}, {5, 6}, {6, 7}, {7, 8}, {8, 9}, {9, 10}}
This is a not partition since it has no subset containing 0 and several elements are
contained in distinct subsets.
2)
* For each of the following relations, state which of these properties hold:
reflexivity, symmetry, transitivity, and antisymmetry.
a)
“equality” defined on strings
This is reflexive(since
(
)
(
)
a
b
a
b
=
⇒
=
), symmetric(since
(
)
(
)
a
b
b
a
=
⇒
=
),
transitive(since
(
)
(
)
a
b
b
c
a
c
=
∧
=
⇒
=
),
and antisymmetric (since
(
)
(
)
a
b
b
a
a
b
=
∧
=
⇒
=
).
b)
“inequality” defined on strings
This is not reflexive (since
(
1
1
)
<
≠<
:
), symmetric (since
(
)
(
)
a
b
b
a
≠
⇒
≠
), not
transitive(since
(
1
2
2
1
)
<
≠<
∧ <
≠<
but
(
1
1
)
<
≠<
:
), and not antisymmetric
(since
(
1
2
2
1
)
<
≠<
∧ <
≠<
but
1
2
<
≠<
).
c)
“less than” defined on
N
(the natural numbers)
This is not reflexive (since
(0
0)
<
:
), , not symmetric (since
(0
1)
<
but
(1
0)
<
:
),
transitive (since
(
)
(
)
a
b
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 Fall '08
 Myers
 Natural number, Total order, partial order, r1

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