csi35chapter 9 revie · 6 csi35chapter 9 review 3. the relation r on the set of all integers is...

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CSI35 Chapter 9 Review

1. Let S be the set of all strings of English letters. Determine whether the following relations are reflexive, irreflexive, symmetric, antisymmetric, asymmetric, and/or transitive.

a) R1 = { (a,b) | a and b have no letters in common}

b) R1 = { (a,b) | a and b are not the same length}

c) R1 = { (a,b) | a is longer than b}

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a) R1 = { (a,b) | a and b have no letters in common}

- irreflexive, not reflexive (“hello”,”hello”) R1

- symmetric (“car”,”top”), (“top”,”car”) R1

- not antisymmertic, not asymmetric- not transitive (“car”,”top”), (“top”,”map”) R

1, but (“car”,”map”) R

1

b) R1 = { (a,b) | a and b are not the same length}


- symmetric (“car”,”memo”), (“memo”,”car”) R1

- not antisymmertic, not asymmetric- not transitive (“car”,”to”), (“to”,”top”) R

1, but (“car”,”top”) R

1

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c) R1 = { (a,b) | a is longer than b}


- not symmetric (“car”,”to”) R1, but (“to”,”car”) R

1

- antisymmertic, asymmetrictransitive (x,y), (y,z) R

1, then (x,z) R

1

of length n

of length mn < m

of length t

m < t

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2. Is relation R on Z Z defined by (a,b) R (c,d) iff a+d = b+c an equivalence relation?

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2. Is relation R on Z Z defined by (a,b) R (c,d) iff a+d = b+c an equivalence relation?

Reflexive? (a,b) R (a,b) ? Yes

Symmetric? If (a,b) R (c,d), does (c,d) R (a,b)? Yes

Transitive? If (a,b) R (c,d) and (c,d) R (f,g), does (a,b) R (f,g)?a+d = b+cc+g = d+f

a+g = b+f? Yes- this is an equivalence relation

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3. The relation R on the set of all integers is defined as: (x,y) R iff xy 0. Determine whether the relations is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, and/or transitive.

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3. The relation R on the set of all integers is defined as: (x,y) R iff xy 0. Determine whether the relations is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, and/or transitive.

Reflexive? Yes (-2,-2) RNot irreflexiveSymmetric? Yes (2,3) and (3,2) RNot antysymmetric, Not asymmetricTransitive? Yes (2,3) and (3,5) R then (2,5) R

(-2,-3) and (-3,-5) R then (-2,-5) RThis is an equivalence relation

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4. Let R be relation on the set {1,2,3,4,5} containing the ordered pairs (1,1), (1,2), (1,3), (2,3), (2,4), (3,1), (3,4), (3,5), (4,1), (4,5), (5,1), (5,2), and (5,4).

Find

a) R-1

b) R

c) R2

d) R3

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Find

a) R-1 = {(1,1), (1,3), (1,4), (1,5), (2,1), (2,5), (3,1), (3,2), (4,2), (4,3), (4,5), (5,3), (5,4)}

b) R = {(1,4), (1,5), (2,1), (2,2), (2,5), (3,2), (3,3), (4,2), (4,3), (4,4), (5,3), (5,5)}

c) R2 = R R = { (1,1),(1,2),(1,3), (1,4), (1,5), (2,1), (2,4), (2,5), (3,1), (3,2), (3,3), (3,4), (3,5), (4,1),(4,2), (4,3), (4,4), (5,1), (5,2), (5,3), (5,4),(5,5)}

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Find

c) R2 = R R = { (1,1),(1,2),(1,3), (1,4), (1,5), (2,1), (2,4), (2,5), (3,1), (3,2), (3,3), (3,4), (3,5), (4,1),(4,2), (4,3), (4,4), (5,1), (5,2), (5,3), (5,4),(5,5)}

d) R3 = R2 R = {1,2,3,4,5} {1,2,3,4,5}

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5. Let R1 and R

2 be relations on the set {a,b,c,d}.

R1 = {(a,b), (b,c), (b,d), (d,a), (d,b)}

R2 = {(a,c), (a,d), (b,c), (b,d), (d,a), (d,c), (d,d)}

Find

a) R1 R

2

b) R1 R

2

c) R1 R

2

d) R1 - R

2

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5. Let R1 and R

2 be relations on the set {a,b,c,d}.

R1 = {(a,b), (b,c), (b,d), (d,a), (d,b)}

R2 = {(a,c), (a,d), (b,c), (b,d), (d,a), (d,c), (d,d)}

Find

a) R1 R

2 = { (b,c), (b,d), (d,a) }

b) R1 R

2 = {(a,b), (a,c), (a,d), (b,c), (b,d), (d,a), (d,c), (d,b),

(d,d)}

c) R1 R

2 = { (a,b), (a,c), (a,d), (d,c), (d,b), (d,d) }

d) R1 - R

2 = {(a,b),(d,b)}

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6. List the ordered pairs in the relation represented by the graph

a b

cd

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6. List the ordered pairs in the relation represented by the graph

{ (a,d), (b,a), (b,d), (c,a), (c,c), (d,b), (d,c), (d,d) }

a b

cd

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7. List the ordered pairs in the relation on {1,2,3,4} represented by the matrix (the integers are listed in increasing order)

[0 1 0 11 0 1 10 1 1 01 0 1 0

]

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7. List the ordered pairs in the relation on {1,2,3,4} represented by the matrix (the integers are listed in increasing order)

[0 1 0 11 0 1 10 1 1 01 0 1 0

]1 2 3 4

1 2 34

{(1,2), (1,4), (2,1), (2,3), (2,4), (3,2), (3,3), (4,1), (4,3)}

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8. Relation R is represented by the matrix M and relation S by the directed graph G. Both relations are on the set of {a,b,c,d}. Determine whether the relations are reflexive, irreflexive, symmetric, antisymmetric, asymmetric, and/or transitive.

[0 1 0 11 0 1 10 1 1 01 0 1 0

]a b

cd

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8. Relation R is represented by the matrix M and relation S by the directed graph G. Both relations are on the set of {a,b,c,d}. Determine whether the relations are reflexive, irreflexive, symmetric, antisymmetric, asymmetric, and/or transitive.

[0 1 0 11 0 1 10 1 1 01 0 1 0

]a b

cd

not reflexive, not irreflexive, not symmetric, not antisymmetric, not asymmetric, not transitive: (a,b), (b,c), but no (a,c)

not reflexive, not irreflexive, not symmetric, not antisymmetric, not asymmetric, not transitive:(d,c),(c,a), but no (d,a)

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Symmetric – antisymmetric – asymmetric

The relations on the set of {1,2,3,4}:

a b

cd

a b

cd

a b

cd

a b

cd

symmetric, antisymmetric, not asymmetric

not symmetric, not asymmetric, antisymmetric

symmetric, not asymmetric, not antisymmetric

not symmetric, asymmetric, antisymmetric

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9. Let R1 and R

2 be relations on a set A represented by the

matrices

M R 1=[0 1 01 1 11 0 0] M R2=[

0 1 00 1 11 1 1]

Find the matrices that representa) R

1 R

2

b) R1 R

2

c) R1 R

2

d) R1 - R

2

e) R1 R

2

f) R1

g) R1

-1

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9. Let R1 and R

2 be relations on a set A represented by the

matrices

M R 1=[0 1 01 1 11 0 0] M R2=[

0 1 00 1 11 1 1]

Find the matrices that represent

e)

g)

M R1∘R2=[0 1 01 1 11 0 0][

0 1 00 1 11 1 1]=[

0 1 11 1 10 1 0]

M R1−1=M R1

T =[0 1 11 1 00 1 0]

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10. Which of these are posets?

a) (R, = ) b) (R, < ) c) (R, ) d) (R, )

i.e. reflexive, antisymmetric, transitive

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10. Which of these are posets?

a) (R, = ) b) (R, < ) c) (R, ) d) (R, )

i.e. reflexive, antisymmetric, transitive

a) (R, = ) poset

b) (R, < ) not reflexive 2 < 2

c) (R, ) poset

d) (R, ) not reflexive 2 = 2

-1.5 -1 3.7 5

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11. Answer the questions for the poset ({2,4,6,9,12,18,27,36,48,60,72}, | ).

a) Draw Hasse diagram of the posetb) Find the maximal elements.c) Find the minimal elements.d) Is there a greatest element?e) Is there a least element?f) Find all upper bounds of { 2, 9 }.g) Find all lower bounds of {60, 72}.

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11. Answer the questions for the poset ({2,4,6,9,12,18,27,36,48,60,72}, | ).

a) Draw Hasse diagram of the posetb) Find the maximal elements.18, 27, 60, 48, 72c) Find the minimal elements.2, 9d) Is there a greatest element?noe) Is there a least element?nof) Find all upper bounds of { 2, 9 }.18, 36, 72g) Find all lower bounds of {60, 72}. 12, 6, 4, 2

48

6

2

4

1218

36

9

27

60

72

csi35chapter 9 revie · 6 csi35chapter 9 review 3. the relation r on the set of all integers is...

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