Question 1

Which of the following are partitions of {1, 2, 3, . . . , 10}?

Accepted Answer

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} 
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} A,D,E

Question 2

suppose that the transactions at a fast-food restaurant during one afternoon are {hamburger, fries, regular soda}, {cheeseburger, fries, regular soda}, {apple, hamburger, fries, regular soda}, {salad, diet soda}, {hamburger, onion rings, regular soda}, {cheeseburger, fries, onion rings, regular soda}, {hamburger, fries}, {hamburger, fries, regular soda}.
-Find the count and support of {cheeseburger}.

Accepted Answer

$\sigma$ ({  cheeseburger })=2 , support  ({ cheeseburger })=  $1 / 4$

Question 3

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.

Accepted Answer

Reflexive: for all $a \in A , a R a$ and $a S a$; hence for all $a \in A , a ( R \cap S ) a$.
Symmetric: suppose $a ( R \cap S ) b$; then $a R b$ and $a S b$; by symmetry of $R$ and $S , b R a$ and bSa; therefore $b ( R \cap S ) a$.
Transitive: suppose $a ( R \cap S ) b$ and $b ( R \cap S ) c$; then $a R b , a S b , b R c$, and $b S c$; by transitivity of $R$ and $S$,$ a R c$ and $a S c$; therefore $a ( R \cap S ) c$.

Question 4

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

Accepted Answer

{(a, b), (a, c), (a,

Question 5

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

Accepted Answer

\[\left( \begin{array} { l l l

Question 6

determine whether the binary relation is:
(1) reﬂexive, (2) symmetric, (3) antisymmetric, (4) transitive.
-The relation R on Z where aR b means $a \neq b$

Accepted Answer

The answer of determine whether the binary relation is:
(1) reﬂexive,...

Question 7

Let $\mathbf { M } _ { R } = \left( \begin{array} { l l l l } 
1 & 0 & 1 & 0 \
1 & 1 & 0 & 1 \
1 & 1 & 1 & 0 \
1 & 1 & 0 & 1
\end{array} \right)$
Determine if  R  is: (a) reflexive, (b) symmetric, (c) antisymmetric, (d) transitive.

Accepted Answer

(a) Yes (b

Question 8

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)}.

Accepted Answer

The answer of determine whether the binary relation is:
(1) reﬂexive,...

Question 9

ﬁnd the matrix that represents the given relation. Use elements in the order given to determine rows and columns of the matrix.
-$$R \text { on } \{ 1,2,4,8,16 \} \text {, where } a R b \text { means } a \mid b \text {. }$$

Accepted Answer

\[\left( \begin{array} { l l l

Question 10

What is the covering relation of the partial ordering $\{ ( a , b ) \quad a \text { divides } b \}$   on the set  {2,4,6,8,10,12} ?

Accepted Answer

{(2, 4), (2, 6), (2,

Question 11

ﬁnd the matrix that represents the given relation. Use elements in the order given to determine rows and columns of the matrix.
-$\bar { R }$, where $R$ is the relation on $\{ w , x , y , z \}$ such that
$$R = \{ ( w , w ) , ( w , x ) , ( x , w ) , ( x , x ) , ( x , z ) , ( y , y ) , ( z , y ) , ( z , z ) \}$$

Accepted Answer

\[\left( \begin{array} { l l l

Question 12

determine whether the binary relation is:
(1) reﬂexive, (2) symmetric, (3) antisymmetric, (4) transitive.
-$$\text { The relation } R \text { on } \mathcal { N } \text { where } a R b \text { means that } a \text { has the same number of digits as } b$$

Accepted Answer

The answer of determine whether the binary relation is:
(1) reﬂexive,...

Question 13

ﬁnd the matrix that represents the given relation. Use elements in the order given to determine rows and columns of the matrix.
-$$R \text { on } \{ w , x , y , z \} , \text { where } R = \{ ( w , w ) , ( w , x ) , ( x , w ) , ( x , x ) , ( x , z ) , ( y , y ) , ( z , y ) , ( z , z ) \}$$

Accepted Answer

\[\left( \begin{array} { l l l

Question 14

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: $$\begin{array} { l c c c } 
& \text { Size Code } & \text { Weight Code } & \text { Shape Code } \
\# 1 & 42 & 27 & 42 \
\# 2 & 27 & 38 & 13 \
\# 3 & 13 & 12 & 27 \
\# 4 & 42 & 38 & 38
\end{array}$$ Find which of the three codes is a primary key. If none of the three codes is a primary key, explain why.

Accepted Answer

The answer of A company makes four kinds of products....

Question 15

What is the covering relation of the partial ordering $\{ ( a , b ) \mid a \text { divides } b \}$ on the set  {1,2,3,4,6,8,12,24} ?

Accepted Answer

{(1, 2), (1, 3), (2,

Question 16

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

Accepted Answer

All except

Question 17

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.

Accepted Answer

The answer of determine whether the binary relation is:
(1) reﬂexive,...

Question 18

Draw the directed graph for the relation defind by the matrix $\left( \begin{array} { l l l l } 
1 & 1 & 1 & 1 \
0 & 1 & 1 & 1 \
0 & 0 & 1 & 1 \
0 & 0 & 0 & 1
\end{array} \right)$

Accepted Answer

The answer of  Draw the directed graph for the...

Question 19

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

Accepted Answer

{(a, a), (

Question 20

suppose R and S are relations on {a, b, c, d}, where $$R = \{ ( a , b ) , ( a , d ) , ( b , c ) , ( c , c ) , ( d , a ) \} \quad \text { and } \quad S = \{ ( a , c ) , ( b , d ) , ( d , a ) \}$$  
Find the combination of relations.
-$$R ^ { 2 }$$

Accepted Answer

{(a, a), (

Exam 9: A: Relations

Which of the following are partitions of {1, 2, 3, . . . , 10}?

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.

determine whether the binary relation is: (1) reﬂexive, (2) symmetric, (3) antisymmetric, (4) transitive. -The relation R on Z where aR b means a≠ba \neq ba=b

Let MR=(1010110111101101)\mathbf { M } _ { R } = \left( \begin{array} { l l l l } 1 & 0 & 1 & 0 \\1 & 1 & 0 & 1 \\1 & 1 & 1 & 0 \\1 & 1 & 0 & 1\end{array} \right)MR​=​1111​0111​1010​0101​​ Determine if R is: (a) reflexive, (b) symmetric, (c) antisymmetric, (d) transitive.

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)}.

What is the covering relation of the partial ordering {(a,b)a divides b}\{ ( a , b ) \quad a \text { divides } b \}{(a,b)a divides b} on the set {2,4,6,8,10,12} ?

What is the covering relation of the partial ordering {(a,b)∣a divides b}\{ ( a , b ) \mid a \text { divides } b \}{(a,b)∣a divides b} on the set {1,2,3,4,6,8,12,24} ?

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

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 directed graph for the relation defind by the matrix (1111011100110001)\left( \begin{array} { l l l l } 1 & 1 & 1 & 1 \\0 & 1 & 1 & 1 \\0 & 0 & 1 & 1 \\0 & 0 & 0 & 1\end{array} \right)​1000​1100​1110​1111​​

The Foundations: Logic and Proofs

A: the Foundations: Logic and Proofs

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

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

Algorithms

A: Algorithms

Number Theory and Cryptography

A: Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: Advanced Counting Techniques

Relations

Graphs

A: Graphs

Trees

A: Trees

Boolean Algebra

A: Boolean Algebra

Modeling Computation

A: Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Filters

determine whether the binary relation is: (1) reﬂexive, (2) symmetric, (3) antisymmetric, (4) transitive. -The relation R on Z where aR b means $a \neq b$

Let $\mathbf { M } _ { R } = \left( \begin{array} { l l l l } 1 & 0 & 1 & 0 \\1 & 1 & 0 & 1 \\1 & 1 & 1 & 0 \\1 & 1 & 0 & 1\end{array} \right)$ Determine if R is: (a) reflexive, (b) symmetric, (c) antisymmetric, (d) transitive.

What is the covering relation of the partial ordering $\{ ( a , b ) \quad a \text { divides } b \}$ on the set {2,4,6,8,10,12} ?

What is the covering relation of the partial ordering $\{ ( a , b ) \mid a \text { divides } b \}$ on the set {1,2,3,4,6,8,12,24} ?

Draw the directed graph for the relation defind by the matrix $\left( \begin{array} { l l l l } 1 & 1 & 1 & 1 \\0 & 1 & 1 & 1 \\0 & 0 & 1 & 1 \\0 & 0 & 0 & 1\end{array} \right)$