Deck 9: Relations

Question 1

List all the binary relations on the set {0, 1}.

Answer 1

Question 2

List the irreflexive relations on the set {0, 1}.

Answer 2

Question 3

In questions determine whether the binary relation is:
(1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive.
The relation R on the set of all people where aR b means that a is at least as tall as b.

Answer 3

1, 4.

Question 4

List the antisymmetric relations on the set {0, 1}.

Question 5

In questions determine whether the binary relation is:
(1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive.
The relation R on {a, b, c} where R = {(a, a), (b, b), (c, c), (a, b), (a, c), (c, b)}.

Question 6

In questions determine whether the binary relation is:
(1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive.
The relation R on Z where aR b means

Question 7

List the relations on the set {0, 1} that are neither reflexive nor irreflexive.

Question 8

In questions determine whether the binary relation is:
(1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive.
The relation R on A = {x, y, z} where R = {(x, x), (y, z), (z, y)}.

Question 9

List the transitive relations on the set {0, 1}.

Question 10

List the symmetric relations on the set {0, 1}.

Question 11

In questions determine whether the binary relation is:
(1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive.
The relation R on {1, 2, 3, . . .} where aR b means a|b.

Question 12

List the relations on the set {0, 1} that are reflexive and symmetric.

Question 13

List the asymmetric relations on the set {0, 1}.

Question 14

List the reflexive relations on the set {0, 1}.

Question 15

In questions determine whether the binary relation is:
(1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive.
The relation R on N where aR b means that a has the same number of digits as b.

Question 16

In questions determine whether the binary relation is:
(1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive.

Question 17

In questions determine whether the binary relation is:
(1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive.
The relation R on Z where aR b means |a − b| ≤ 1.

Question 18

In questions determine whether the binary relation is:
(1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive.
The relation R on Z where aR b means

Question 19

In questions determine whether the binary relation is:
(1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive.
The relation R on Z where aR b means that the units digit of a is equal to the units digit of b.

Question 20

In questions determine whether the binary relation is:
(1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive.
The relation R on {w, x, y, z} where R = {(w, w), (w, x), (x, w), (x, x), (x, z), (y, y), (z, y), (z, z)}.

Question 21

In questions find the matrix that represents the given relation. Use elements in the order given to
determine rows and columns of the matrix.

Question 22

In questions find the matrix that represents the given relation. Use elements in the order given to
determine rows and columns of the matrix.

Question 23

In questions find the matrix that represents the given relation. Use elements in the order given to
determine rows and columns of the matrix.

Question 24

A company makes four kinds of products. Each product has a size code, a weight code, and a shape code.
The following table shows these codes: Find which of the three codes is a primary key. If none of the three codes is a primary key, explain why.

Question 25

In questions find the matrix that represents the given relation. Use elements in the order given to
determine rows and columns of the matrix.

Question 26

In questions find the matrix that represents the given relation. Use elements in the order given to
determine rows and columns of the matrix.

Question 27

supp ose R and S are relations on {a, b, c, d}, where
Find the combination of relations.

Question 28

If X =(Fran Williams, 617885197, MTH 202, 248B West), find the projections

Question 29

In questions find the matrix that represents the given relation. Use elements in the order given to
determine rows and columns of the matrix.

Question 30

In questions find the matrix that represents the given relation. Use elements in the order given to
determine rows and columns of the matrix.

Question 31

In questions determine whether the binary relation is:
(1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive.

Question 32

supp ose R and S are relations on {a, b, c, d}, where
Find the combination of relations.

Question 33

supp ose R and S are relations on {a, b, c, d}, where
Find the combination of relations.

Question 34

In questions determine whether the binary relation is:
(1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive.

Question 35

supp ose R and S are relations on {a, b, c, d}, where
Find the combination of relations.

Question 36

In questions find the matrix that represents the given relation. Use elements in the order given to
determine rows and columns of the matrix.

Question 37

supp ose R and S are relations on {a, b, c, d}, where
Find the combination of relations.

Question 38

In questions find the matrix that represents the given relation. Use elements in the order given to
determine rows and columns of the matrix.

Question 39

In questions determine whether the binary relation is:
(1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive.
The relation R on the set of all people where aR b means that a is younger than b.

Question 40

supp ose R and S are relations on {a, b, c, d}, where
Find the combination of relations.

Question 41

If R = {(1, 2), (1, 4), (2, 3), (3, 1), (4, 2)}, find the symmetric closure of R.

Question 42

What is the covering relation of the partial ordering

Question 43

Draw the Hasse diagram for the relation R on A = {2, 3, 4, 5, 6, 8, 10, 40}

Question 44

Find the transitive closure of R if is

Question 45

Find the smallest equivalence relation on {1, 2, 3} that contains (1, 2) and (2, 3).

Question 46

Find the transitive closure of R if

Question 47

Suppose A={2,3,6,9,10,12,14,18,20} and R is the partial order relation defined on A where xRy means x is a divisor of y .
(a) Draw the Hasse diagram for R .
(b) Find all maximal elements.
(c) Find all minimal elements.
(d) Find lub({2,9}) .
(e) Find lub ({3,10}) .
(f) Find glb ({14,10}) .

Question 48

Find the join of the 3 -ary relation
{ (Wages,MS410,N507 ),(Rosen, CS540,N525), (Michaels, CS518,N504), (Michaels,MS410,N510)}
and the 4 -ary relation
{(MS410,N507,Monday, 6: 00),(MS410,N507,Wednesday, 6: 00),(CS540,N525, Monday,7:30),
(CS518,N504,Tuesday,6:00), (CS518,N504,Thursday,6:00) }
with respect to the last two fields of the first relation and the first two fields of the second relation.

Question 49

Find the transitive closure of R on {a, b, c, d} where R={(a, a),(b, a),(b, c),(c, a),(c, c),(c, d),(d, a),(d, c)} .

Question 50

Draw the directed graph for the relation defined by the matrix

Question 51

If R = {(1, 2), (1, 4), (2, 3), (3, 1), (4, 2)}, find the reflexive closure of R.

Question 52

Draw the directed graph for the relation defined by the matrix

Question 53

Find the smallest partial order relation on {1, 2, 3} that contains (1, 1), (3, 2), (1, 3).

Question 54

Draw the Hasse diagram for the relation R on A = {2, 3, 4, 6, 10, 12, 16}

Question 55

The diagram at the right is the Hasse diagram for a partially ordered set. Referring to this diagram:
(a) List the maximal elements
(b) List the minimal elements
(c) Find all upper bounds for f, g
(d) Find all lower bounds for d, f
(e) Find lub({g, j, m})
(f) Find glb({d, e})
(g) Find the greatest element
(h) Find the least element
(i) Use a topological sort to order the elements of the poset represented by this Hasse diagram.

Question 56

If determine if R is:
(a) reflexive
(b) symmetric
(c) antisymmetric
(d) transitive.

Question 57

If R= and are bit strings containing the same number of 0s}, find the equivalence classes of
(a) 1 .
(b) 00 .
(c) 101 .

Question 58

If determine if R is:
(a) reflexive
(b) symmetric
(c) antisymmetric
(d) transitive.

Question 59

What is the covering relation of the partial ordering

Question 60

In questions find the matrix that represents the given relation. Use elements in the order given to
determine rows and columns of the matrix.

Question 61

In questions give an example or else prove that there are none.
A relation on {1, 2, 3} that is reflexive and transitive, but not symmetric.

Question 62

Let R be the relation on A={1,2,3,4,5} where R={(1,1),(1,3),(1,4),(2,2),(3,1),(3,3),(3,4),(4,1), , (4,3),(4,4),(5,5)} . Draw the directed graph for R .

Question 63

Suppwhetherose RR isis thean equivrelationalenceonrelationN whereonaRNb. means that a ends in the same digit in which b ends. Determine

Question 64

Suppose that R and S are equivalence relations on a set A . Prove that the relation R∩S is also an equivalence relation on A .

Question 65

Let R be the relation on A={1,2,3,4,5} where R={(1,1),(1,3),(1,4),(2,2),(3,1),(3,3),(3,4),(4,1) , (4,3),(4,4),(5,5)} . R is an equivalence relation. Find the equivalence classes for the partition of A given by R .

Question 66

Which of the following are partitions of {1,2,3, ... , 10} ?
(a) {2,4,6,8},{1,3,5,9},{7,10} .
(b) {1,2,4,8},{2,5,7,10},{3,6,9} .
(c) {3,8,10},{1,2,5,9},{4,7,8} .
(d) {1},{2}, ... ,{10} .
(e) {1,2, ... , 10} .

Question 67

Suppose A is the set composed of all ordered pairs of positive integers. Let R be the relation defined on A where (a, b) R(c, d) means that a+d=b+c .
(a) Prove that R is an equivalence relation.
(b) Find [(2,4)] .

Question 68

Suppose Find the number of reflexive, symmetric binary relations on A .

Question 69

In questions give an example or else prove that there are none.

Question 70

Suppose . Find the number of binary relations on A .

Question 71

In questions give an example or else prove that there are none.
A relation on {1, 2} that is symmetric and transitive, but not reflexive.

Question 72

Let R be the relation on A={1,2,3,4,5} where R={(1,1),(1,3),(1,4),(2,2),(3,1),(3,3),(3,4),(4,1) , (4,3),(4,4),(5,5)} . Write the matrix for R .

Question 73

Suppose Find the number of symmetric binary relations on A .

Question 74

Suppose the relation R is defined on the set where aRb means that Determine whether R is an equivalence relation on