1
Chapter 2 Relations – Practice Problem Set IV
Figure 1
1. [BRK 4.3 #1-8] Refer to figure 1 which represents the directed graph (digraph) of the relation
€
R on
€
A = 1,2,3,4,5,6{ }.
a) List all paths of length 1. b) List all paths of length 2 starting from vertex 2. c) List all paths of length 2. d) List all paths of length 3 starting from vertex 3. e) List all paths of length 3 (one can see that it is more difficult to find paths of
length 3 than paths of length 2). f) Find a cycle starting at vertex 2. g) Find a cycle starting at vertex 6. h) Draw the digraph of
€
R2. i) Find
€
MR 2 . j) Find
€
R∞ (using directed graph). k) Find
€
MR∞ .
2
1
5
3
4
6
2
Figure 2
2. [BRK 4.3 #24] Refer to figure 2 which represents the directed graph (digraph) of the
relation
€
R on
€
A = 1,2,3,4,5,6,7{ }. Find two cycles of length at least 3 in the relation
€
R. 3. [BRK 4.3 #26] Let
€
A = 1,2,3,4,5{ } and
€
R be the relation defined by
€
aRb if and only if
€
a < b. a) Compute
€
R2 and
€
R3. b) Complete the following statement:
€
aR2b if and only if _______________.
c) Complete the following statement:
€
aR3b if and only if _______________.
4. [2.9] Consider the following five relations on the set
€
A = 1,2,3{ }.
€
R = 1,1( ), 1,2( ), 1,3( ), 3,3( ){ }
€
∅ =empty relation
€
S = 1,1( ), 1,2( ), 2,1( ), 2,2( ), 3,3( ){ }
€
A × A =universal relation
€
T = 1,1( ), 1,2( ), 2,2( ), 2,3( ){ } Determine whether or not each of the five relations on
€
A is: a) reflexive b) symmetric c) transitive
2
1
5
3 4
7
6
3
For the next three problems, determine whether or not each relation on
€
A is: a) reflexive b) symmetric c) transitive
5. [BRK 4.4 #1-4, #7-8] Let
€
A = 1,2,3,4{ } and consider the following relations on
€
A .
a)
€
R = 1,1( ), 1,2( ), 2,1( ), 2,2( ), 3,3( ), 3,4( ), 4,3( ), 4,4( ){ } b)
€
R = 1,2( ), 1,3( ), 1,4( ), 2,3( ), 2,4( ), 3,4( ){ } c)
€
R = 1,3( ), 1,1( ), 3,1( ), 1,2( ), 3,3( ), 4,4( ){ } d)
€
R = 1,1( ), 2,2( ), 3,3( ){ } e)
€
R = 1,2( ), 1,3( ), 3,1( ), 1,1( ), 3,3( ), 3,2( ), 1,4( ), 4,2( ), 3,4( ){ } f)
€
R = 1,3( ), 4,2( ), 2,4( ), 3,1( ), 2,2( ){ }
6. [BRK 4.4 #9-10] Let
€
A = 1,2,3,4,5{ } and consider the following relations on
€
A represented by the directed graphs shown in figures 3 and 4.
Figure 3 Figure 4
7. [BRK 4.4 #11-12] Let
€
A = 1,2,3,4{ } and consider the following relation
€
R on
€
A represented by the each matrix of the relation.
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
0011001011011010
RM
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
1000010000110011
RM
2 1
3
4
5
2 1
5
3
4
4
For the next three problems, determine whether or not each relation on
€
A is an equivalence relation.
8. [BRK 4.5 #1-2] Let
€
A = a,b,c{ } and consider the following relation
€
R on
€
A represented by the each matrix of the relation.
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=
110110001
RM ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=
100010101
RM
9. [BRK 4.5 #4] Let
€
A = 1,2,3{ } and consider the following relation on
€
A represented by the directed graph shown in figure 5.
Figure 5 10. [BRK 4.5 #5-7] Consider the following relations on
€
A .
a)
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A = a,b,c,d{ },R = a,a( ), b,a( ), b.b( ), c,c( ), d,c( ), d,d( ){ } b)
€
A = 1,2,3,4,5{ },R = 1,1( ), 1,2( ), 1,3( ), 2,1( ), 2,2( ), 2,3( ), 3,1( ), 3,2( ), 3,3( ), 4,4( ), 5,5( ){ } c)
€
A = 1,2,3,4{ },R = 1,1( ), 1,2( ), 1,3( ), 2,1( ), 2,2( ), 3,1( ), 3,3( ), 4,1( ), 4,4( ){ }
2 1
3
5
11. [2.7] Let
€
R and
€
S be the following relations on
€
A = 1,2,3{ }:
€
R = 1,1( ), 1,2( ), 2,3( ), 3,1( ), 3,3( ){ } and
€
S = 1,2( ), 1,3( ), 2,1( ), 3,3( ){ }
Find: a)
€
R∪ S b)
€
R∩ S c)
€
RC or
€
R
12. [BRK 4.7 #1] Let
€
A = 1,2,3{ }. Let
€
R and
€
S be relations on
€
A given as
€
R = 1,1( ), 1,2( ), 2,3( ), 3,1( ){ } and
€
S = 2,1( ), 3,1( ), 3,2( ), 3,3( ){ }. Compute: a)
€
R b)
€
R∩ S c)
€
R∪ S d)
€
S−1
13. [BRK 4.7 #2] Let
€
A = a,b,c{ } and
€
B = 1,2,3{ } . Let
€
R and
€
S be relations from
€
A to
€
B given as
€
R = a,1( ), b,1( ), c,2( ), c,3( ){ } and
€
S = a,1( ), a,2( ), b,1( ), b,2( ){ } . Compute: a)
€
R b)
€
R∩ S c)
€
R∪ S d)
€
S−1
6
14. [BRK 4.7 #11-12] Let
€
A = 1,2,3,4{ } and
€
B = 1,2,3{ } . Let
€
R and
€
S be relations from
€
A to
€
B given by the matrices below. For each pair of matrices below, compute:
a)
€
MR∩S b)
€
MR∪S c)
€
MR −1 d)
€
MS
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
111101101010
,
101010110101
SR MM
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
010010101101
,
111100110010
SR MM
15. [BRK 4.7 #16] Let
€
A = 1,2,3,4{ } and
€
R = 2,1( ), 2,2( ), 2,3( ), 3,2( ), 3,3( ), 4,2( ){ }.
a) Find the reflexive closure of
€
R. b) Find the symmetric closure of
€
R.
16. [BRK 4.7 #17] Let
€
R be the relation whose matrix is
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
1010000110001111010011001
a) Find the reflexive closure of
€
R. b) Find the symmetric closure of
€
R.
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17. [2.13] Consider the relation
€
R = a,a( ), a,b( ), b,c( ), c,c( ){ } on the set
€
A = a,b,c{ }. Find the reflexive, symmetric, and transitive closures of
€
R:
a) reflexive
€
R( ) b) symmetric
€
R( ) c) transitive
€
R( )