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4.1
/1, 810, 20, 21
1.
For each of the following binary relations
ρ
on
, decide which of the given ordered pairs belong to
ρ
.
a.
x
ρ
y
↔
x + y <
7;
(1, 3),
(2, 5),
(3, 3),
(4, 4)
b.
x
ρ
y
↔
x = y +
2;
(0, 2),
(4, 2),
(6, 3),
(5, 3)
c.
x
ρ
y
↔
2
x
+ 3
y
= 10;
(5, 0),
(2, 2),
(3, 1),
(1, 3)
d.
x
ρ
y
↔
y
is a perfect square; (1, 1),
(4, 2),
(3, 9),
(25, 5)
8.
Let
S=
{0, 1, 2, 4, 6}. Test the following binary relations on
S
for reflexivity, symmetry, antisymmetry, and
transitivity.
a.
ρ
=
{(0, 0), (1, 1), (2, 2), (4, 4), (6, 6), (0, 1), (1, 2), (2, 4), (4, 6)}
reflexive, antisymmetric
b.
ρ
=
{(0, 1), (1, 0), (2, 4), (4, 2), (4, 6), (6, 4)}
symmetric
c.
ρ
=
{(0, 1), (1, 2), (0, 2), (2, 0), (2, 1), (1, 0), (0, 0), (1, 1), (2, 2)}
symmetric, transitive
d.
ρ
=
{(0, 0), (1, 1), (2, 2), (4, 4), (6, 6), (4, 6), (6, 4)}
reflexive, symmetric, transitive
e.
ρ
=
{}
symmetric, antisymmetric, transitive
9.
Let
S
be the set of people in the United States. Test the following binary relations on
S
for reflexivity,
symmetry, antisymmetry, and transitivity.
a
. x
ρ
y
↔
x
is at least as tall as
y
.
reflexive, antisymmetric, transitive
b.
x
ρ
y
↔
x
is taller than
y
.
antisymmetric, transitive
c.
x
ρ
y
↔
x
is the same height as
y
.
reflexive, symmetric, transitive
d.
x
ρ
y
↔
x
is a child of
y
.
antisymmetric
e.
x
ρ
y
↔
x
is the husband of
y
.
antisymmetric, transitive.
If x is the husband of y then y is
assumed to be a woman, so we never have
x
ρ
y
and
y
ρ
x
except when
x = y
f.
x
ρ
y
↔
x
is the spouse of
y
.
symmetric
g.
x
ρ
y
↔
x
has the same parents as y.
reflexive, symmetric, transitive
h.
x
ρ
y
↔
x
is the brother of y.
none. not symmetric because
y
could be the sister
of
x
.
10.
Test the following binary relations on the given sets
S
for reflexivity, symmetry, antisymmetry, and
transitivity.
a.
S=
,
x
ρ
y
↔
x
≤

y
.
reflexive, transitive.
(not antisymmetric: 1
≤
1, 1
≤
1)
b.
S=
, x
ρ
y
↔
x – y
is
an integral multiple of 3
reflexive, symmetric, transitive
c.
S=
, x
ρ
y
↔
x
•
y
is even
symmetric.
(not transitive:
3
ρ
8
and
8
ρ
5,
but
3
not
ρ
5
)
d.
S=
,
x
ρ
y
↔
x
is
odd
transitive
e.
S=
set of all squares in the plane,
S
1
ρ
S
2
↔
length of side of
S
1
=
length of side of
S
2
reflexive, symmetric, transitive.
f.
S=
set of all finitelength strings of characters,
x
ρ
y
↔
number of characters in x = number of characters
in
y
reflexive, symmetric, transitive.
g.
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 Spring '10
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