Exam 9: Relations

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} .

(Short Answer)

5.0/5

(64)

Question 61

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.

(Short Answer)

4.8/5

(82)

Question 62

Draw the Hasse diagram for the relation R on A = {2, 3, 4, 6, 10, 12, 16} $\text { where } a R b \text { means } a \mid b \text {. }$

(Short Answer)

4.9/5

(96)

Question 63

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

(Short Answer)

4.8/5

(91)

Question 64

supp $R = \{ ( a , b ) , ( a , d ) , ( b , c ) , ( c , c ) , ( d , a ) \} \text { and } S = \{ ( a , c ) , ( b , d ) , ( d , a ) \}$ ose R and S are relations on {a, b, c, d}, where Find the combination of relations. - $S \circ R$

(Short Answer)

4.9/5

(68)

Question 65

supp $R = \{ ( a , b ) , ( a , d ) , ( b , c ) , ( c , c ) , ( d , a ) \} \text { and } S = \{ ( a , c ) , ( b , d ) , ( d , a ) \}$ ose R and S are relations on {a, b, c, d}, where Find the combination of relations. - $S ^ { 3 }$

(Short Answer)

4.8/5

(87)

Question 66

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.

(Short Answer)

4.8/5

(86)

Question 67

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

(Short Answer)

4.8/5

(89)

Question 68

Find the transitive closure of R if $\mathbf { M } _ { R }$ is $\left( \begin{array} { l l l } 1 & 0 & 0 \\0 & 1 & 1 \\1 & 0 & 1\end{array} \right)$

(Short Answer)

4.9/5

(64)

Question 69

In questions ﬁnd the matrix that represents the given relation. Use elements in the order given to determine rows and columns of the matrix. - $R ^ { 2 } \text {, where } R \text { is the relation on on } \{ 1,2,3,4 \} \text { such that } a R b \text { means } | a - b | \leq 1 \text {. }$

(Short Answer)

4.9/5

(95)

Question 70

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

(Short Answer)

4.8/5

(71)

Question 71

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.

(Short Answer)

4.9/5

(80)

Question 72

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.

(Short Answer)

4.7/5

(99)

Question 73

Draw the Hasse diagram for the relation R on A = {2, 3, 4, 5, 6, 8, 10, 40} $\text { where } a R b \text { means } a \mid b \text {. }$

(Short Answer)

4.8/5

(77)

Question 74

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} .

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.

Draw the Hasse diagram for the relation R on A = {2, 3, 4, 6, 10, 12, 16} $\text { where } a R b \text { means } a \mid b \text {. }$

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

supp $R = \{ ( a , b ) , ( a , d ) , ( b , c ) , ( c , c ) , ( d , a ) \} \text { and } S = \{ ( a , c ) , ( b , d ) , ( d , a ) \}$ ose R and S are relations on {a, b, c, d}, where Find the combination of relations. - $S \circ R$

supp $R = \{ ( a , b ) , ( a , d ) , ( b , c ) , ( c , c ) , ( d , a ) \} \text { and } S = \{ ( a , c ) , ( b , d ) , ( d , a ) \}$ ose R and S are relations on {a, b, c, d}, where Find the combination of relations. - $S ^ { 3 }$

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.

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

Find the transitive closure of R if $\mathbf { M } _ { R }$ is $\left( \begin{array} { l l l } 1 & 0 & 0 \\0 & 1 & 1 \\1 & 0 & 1\end{array} \right)$

In questions ﬁnd the matrix that represents the given relation. Use elements in the order given to determine rows and columns of the matrix. - $R ^ { 2 } \text {, where } R \text { is the relation on on } \{ 1,2,3,4 \} \text { such that } a R b \text { means } | a - b | \leq 1 \text {. }$

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

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.

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.

Draw the Hasse diagram for the relation R on A = {2, 3, 4, 5, 6, 8, 10, 40} $\text { where } a R b \text { means } a \mid b \text {. }$

The Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

Number Theory and Cryptography

Induction and Recursion

Counting

Discrete Probability

Advanced Counting Techniques

Graphs

Trees

Boolean Algebra

Modeling Computation

Filters

Exam 9: Relations

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} .

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.

Draw the Hasse diagram for the relation R on A = {2, 3, 4, 6, 10, 12, 16} where aRb means a∣b. \text { where } a R b \text { means } a \mid b \text {. } where aRb means a∣b.

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

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.

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

Find the transitive closure of R if MR\mathbf { M } _ { R }MR​ is (100011101)\left( \begin{array} { l l l } 1 & 0 & 0 \\0 & 1 & 1 \\1 & 0 & 1\end{array} \right)​101​010​011​​

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

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.

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.

Draw the Hasse diagram for the relation R on A = {2, 3, 4, 5, 6, 8, 10, 40} where aRb means a∣b. \text { where } a R b \text { means } a \mid b \text {. } where aRb means a∣b.

The Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

Number Theory and Cryptography

Induction and Recursion

Counting

Discrete Probability

Advanced Counting Techniques

Graphs

Trees

Boolean Algebra

Modeling Computation

Filters

Draw the Hasse diagram for the relation R on A = {2, 3, 4, 6, 10, 12, 16} $\text { where } a R b \text { means } a \mid b \text {. }$

Find the transitive closure of R if $\mathbf { M } _ { R }$ is $\left( \begin{array} { l l l } 1 & 0 & 0 \\0 & 1 & 1 \\1 & 0 & 1\end{array} \right)$

Draw the Hasse diagram for the relation R on A = {2, 3, 4, 5, 6, 8, 10, 40} $\text { where } a R b \text { means } a \mid b \text {. }$