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

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R∞ (using directed graph). k) Find

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MR∞ .

2

1

5

3

4

6

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

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A = 1,2,3,4,5{ } and

€

R be the relation defined by

€

aRb if and only if

€

a < b. a) Compute

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R2 and

€

R3. b) Complete the following statement:

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aR2b if and only if _______________.

c) Complete the following statement:

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aR3b if and only if _______________.

4. [2.9] Consider the following five relations on the set

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A = 1,2,3{ }.

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R = 1,1( ), 1,2( ), 1,3( ), 3,3( ){ }

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∅ =empty relation

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S = 1,1( ), 1,2( ), 2,1( ), 2,2( ), 3,3( ){ }

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A × A =universal relation

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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

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)

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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)

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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

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)

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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)

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A = 1,2,3,4{ },R = 1,1( ), 1,2( ), 1,3( ), 2,1( ), 2,2( ), 3,1( ), 3,3( ), 4,1( ), 4,4( ){ }

11. [2.7] Let

€

R and

€

S be the following relations on

€

A = 1,2,3{ }:

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R = 1,1( ), 1,2( ), 2,3( ), 3,1( ), 3,3( ){ } and

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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

2 1

3

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 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)

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MR −1 d)

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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.

17. [2.13] Consider the relation

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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( )