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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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}
- 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}
- irreflexive, not reflexive (“hello”,”hello”) R1
- 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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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.
c) R1 = { (a,b) | a is longer than b}
- irreflexive, not reflexive (“hello”,”hello”) R1
- not symmetric (“car”,”to”) R1, but (“to”,”car”) R
1
- antisymmertic, asymmetric- transitive (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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CSI35 Chapter 9 Review
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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CSI35 Chapter 9 Review
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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CSI35 Chapter 9 Review
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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CSI35 Chapter 9 Review
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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CSI35 Chapter 9 Review
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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CSI35 Chapter 9 Review
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 = {(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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CSI35 Chapter 9 Review
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
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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CSI35 Chapter 9 Review
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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CSI35 Chapter 9 Review
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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CSI35 Chapter 9 Review
6. List the ordered pairs in the relation represented by the graph
a b
cd
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CSI35 Chapter 9 Review
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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CSI35 Chapter 9 Review
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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CSI35 Chapter 9 Review
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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CSI35 Chapter 9 Review
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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CSI35 Chapter 9 Review
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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CSI35 Chapter 9 Review
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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CSI35 Chapter 9 Review
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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CSI35 Chapter 9 Review
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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CSI35 Chapter 9 Review
10. Which of these are posets?
a) (R, = ) b) (R, < ) c) (R, ) d) (R, )
i.e. reflexive, antisymmetric, transitive
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CSI35 Chapter 9 Review
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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CSI35 Chapter 9 Review
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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CSI35 Chapter 9 Review
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
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