COT3100
Spring’2001
Practice problems for the test#2
Relations.
1.
Let
A
={1, 2, 3},
B
={
w
,
x
,
y
,
z
}, and
C
= {4, 5, 6}. Define the relation
R
⊂
A
×
B
,
S
⊂
B
×
C
, and
T
⊂
B
×
C
, where
R
= {(1,
w
), (3,
w
), (2,
x
), (1,
y
)},
S
={(
w
, 5), (
x
, 6), (
y
, 4),
(
y
, 6)}, and
T
={(
w
, 4), (
w
, 5), (
y
, 5)}.
a)
Determine
R
(
S
∪
T
) and (
R
S
)
∪
(
R
T
).
b)
Determine
R
(
S
∩
T
) and (
R
S
)
∩
(
R
T
).
2.
Consider the relation
T
over real numbers:
T
= {(
a
,
b
) 
a
2
+
b
2
=1}. Define all
properties of this relation (reflexive, irreflexive, symmetric, antisymmetric, and
transitive).
3.
Let
R
⊂
A
×
A
be a symmetric relation. Prove that
t
(
R
) is symmetric.
4.
Prove or disprove the following propositions about arbitrary relations on
A
:
a)
R
1
,
R
2
symmetric
→
R
1
∪
R
2
is symmetric.
b)
R
1
,
R
2
transitive
→
R
1
∩
R
2
is transitive.
5.
Prove or disprove that for any relation
R
⊆
A
×
A
ts
(
R
)
⊆
st
(
R
).
6.
Let
R
be a symmetric and transitive relation on set
S
. Furthermore, suppose, that for
every
x
∈
S
there is an element
y
∈
S
such that
xRy
. Prove that
R
is equivalence relation.
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 Spring '09
 Binary relation, Transitive relation, Symmetric relation, relation, Reflexive relation

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