MAS 111 FOUNDATION OF MATHEMATICS
ASSIGNMENT 10
HINTS
1. Here are some relations on a set
A
=
{
1
,
2
,
3
,
4
,
5
}
. Determine for each
relation whether it is reﬂexive, whether it is transitive, whether is it
antisymmetric. Draw the directed graph that represents each relation.
(a)
R
=
{
(1
,
1)
,
(2
,
2)
,
(2
,
3)
,
(3
,
2)
,
(3
,
3)
,
(3
,
4)
,
(4
,
3)
,
(4
,
4)
,
(5
,
5)
}
.
(b)
S
=
{
(1
,
1)
,
(1
,
2)
,
(1
,
4)
,
(1
,
5)
,
(2
,
2)
,
(2
,
4)
,
(2
,
5)
,
(3
,
3)
,
(3
,
4)
,
(3
,
5)
,
(4
,
4)
,
(4
,
5)
,
(5
,
5)
}
.
(c)
T
=
{
(1
,
3)
,
(1
,
5)
,
(2
,
4)
,
(3
,
1)
,
(3
,
5)
,
(4
,
2)
,
(5
,
1)
,
(5
,
3)
}
.
(d)
U
=
{
(1
,
1)
,
(1
,
3)
,
(1
,
5)
,
(2
,
2)
,
(2
,
4)
,
(3
,
1)
,
(3
,
3)
,
(3
,
5)
,
(4
,
2)
,
(4
,
4)
,
(5
,
1)
,
(5
,
3)
,
(5
,
5)
}
.
Hints
: To solve such questions, ﬁrst draw the directed graphs (arrow
diagrams) and then decide whether they are reﬂexive, or transitive or
antisymmetric.
(a)
R
=
{
(1
,
1)
,
(2
,
2)
,
(2
,
3)
,
(3
,
2)
,
(3
,
3)
,
(3
,
4)
,
(4
,
3)
,
(4
,
4)
,
(5
,
5)
}
.
r
r
r
r
r
2
3
5
1
4

±

±
Each of element has a loop. Please add these loops when you read
it.
R
is reﬂexive, symmetric, but not transitive ((2
,
3)
,
(3
,
4) are in
R
but (2
,
4) is not), not antisymmetric (since both (2
,
3) and (3
,
2)
are in
R
).
(b)
S
=
{
(1
,
1)
,
(1
,
2)
,
(1
,
4)
,
(1
,
5)
,
(2
,
2)
,
(2
,
4)
,
(2
,
5)
,
(3
,
3)
,
(3
,
4)
,
(3
,
5)
,
(4
,
4)
,
(4
,
5)
,
(5
,
5)
}
1