Question 1

$\text { Use the fact that } 29 = 16 + 8 + 4 + 1 \text { to compute } 18 ^ { 29 } \bmod 65 \text {. }$

Accepted Answer

$18 ^ { 1 } \bmod 65 = 18$
$18 ^ { 2 } \bmod 65 = 324 \bmod 65 = 64 \bmod 65$
$18 ^ { 4 } \bmod 65 = 64 ^ { 2 } \bmod 65 = 4096 \bmod 65 = 1$
$18 ^ { 8 } \bmod 65 = 1 ^ { 2 } \bmod 65 = 1$
$18 ^ { 16 } \bmod 65 = 1 ^ { 2 } \bmod 65 = 1$
Hence, by Theorem 8.4.3,
$18 ^ { 29 } = 18 ^ { 16 + 8 + 4 + 1 } = 18 ^ { 16 } 18 ^ { 8 } 18 ^ { 4 } 18 ^ { 1 } \equiv 1 \cdot 1 \cdot 1 \cdot 18 \equiv 18 ( \bmod 65 )$ $\text { and thus } 18 ^ { 29 } \bmod 65 = 18$

Question 2

Let $A = \{ 2,3,4,5,6,7,8 \}$ and define a relation $R$ on $A$ as follows: for all $x , y \in A$,
$x R y \Leftrightarrow 3 \mid ( x - y ) .$
(a) Is $7 R 2$ ? Is $13 R 4$ ? Is $2 R 5$ ? Is $8 R 8$ ?
(b) Draw the directed graph of $R$.

Accepted Answer

a. $7 R 2$ because $7 - 2 = 5$ and $3 \nmid 5 ; \quad 13 R 4$ because $13 - 4 = 9$ and $3 \mid 9$;
$2 R 5$ because $2 - 5 = - 3$ and $3 \mid - 3 ; \quad 8 R 8$ because $8 - 8 = 0$ and $3 \mid 0$.
b. Draw the directed graph of $R$.

Question 3

Let $A = \{ 0,1,2,3 \}$ and define a relation $R$ on $A$ as follows: $R = \{ ( 0,2 ) , ( 0,3 ) , ( 2,0 ) , ( 2,1 ) \}$.
(a) Draw the directed graph of $R$.
(b) Is $R$ reflexive? Explain.
(c) Is $R$ symmetric? Explain.
(d) Is $R$ transitive? Explain.

Accepted Answer

a.   b. $R$ is not reflexive because, for example, $( 1,1 ) \notin R$.
c. $R$ is not symmetric because, for example, $( 0,3 ) \in R$ but $( 3,0 ) \notin R$.
d. $R$ is not transitive because, for example, $( 0,2 ) \in R$ and $( 2,0 ) \in R$ but $( 0,0 ) \notin R$.

Question 4

Let $B = \{ 0,1,2,3 \}$ and define a relation $U$ on $B$ by
$$U = \{ ( 0,2 ) , ( 0,3 ) , ( 2,0 ) , ( 2,1 ) \}$$
Is $U$ transitive? Justify your answer.

Accepted Answer

The answer of Let $B = \{ 0,1,2,3 \}$ and...

Question 5

Define a relation $R$ from $\{ a , b , c \}$ to $\{ u , v \}$ as follows: $R = \{ ( a , u ) , ( b , u ) , ( c , v ) \}$.
(a) Draw an arrow diagram for $R$.
(b) Is $R$ a function? Why or why not?

Accepted Answer

a. @#IMG-DLM& b. R is a function because

Question 6

An RSA cipher has public key pq = 65 and e = 7.
(a) Translate the message YES into its numeric equivalent, and use the formula $C = M ^ { e }$  (mod pq) to encrypt the message.
(b) Decrypt the ciphertext 50 16 and translate the result into letters of the alphabet to discover the message.

Accepted Answer

a. The numeric equivalents of @#LAT-DLM&, and @#LAT-DLM& are

Question 7

Define a relation $R$ from $\{ a , b , c \}$ to $\{ u , v \}$ as follows: $R = \{ ( a , v ) , ( b , u ) , ( b , v ) , ( c , u ) \}$.
(a) Draw an arrow diagram for $R$.
(b) Is $R$ a function? Why or why not?

Accepted Answer

a. @#IMG-DLM& b.R is not a function beca

Question 8

Deﬁne a relation S on the set of positive integers as follows: for all positive integers m and n, $$m S n \Leftrightarrow m \mid n$$ 
(a) Is S reﬂexive? Justify your answer.
(b) Is S symmetric? Justify your answer.

Accepted Answer

a. @#LAT-DLM& is reflexive because for e

Question 9

Let $A = \{ 3,4,5,6,7 \}$ and define a relation $R$ on $A$ as follows: for all $x , y \in A$,
$$x R y \Leftrightarrow 2 \mid ( x - y ) .$$ 
(a) Is 6 R 3? Is 4 R 6?
(b) Draw the directed graph of R.

Accepted Answer

a. @#LAT-DLM& 
b. Dr

Question 10

Let R be the relation deﬁned on the set of all integers Z as follows: for all integers m and n, $$m R n \Longleftrightarrow m - n \text { is divisible by } 5 \text {. }$$ 
(a) Is R reﬂexive? Prove or give a counterexample.
(b) Is R symmetric? Prove or give a counterexample.
(c) Is R transitive? Prove or give a counterexample.

Accepted Answer

a. @#LAT-DLM& is reflexive. Proof: Suppose @#LAT-DLM& is any

Question 11

Define a relation $T$ on $\mathbf { R }$ as follows: for all $x$ and $y$ in $\mathbf { R } , x T y$ if and only if $x ^ { 2 } = y ^ { 2 }$. Then $T$ is an equivalence relation on $\mathbf { R }$. 
(a) Prove that T is an equivalence relation on R.
(b) Find the distinct equivalence classes of T.

Accepted Answer

a. Proof:
 @#LAT-DLM& is reflexive because for eac

Question 12

Let $B = \{ 0,1,2,3 \}$ and define a relation $U$ on $B$ by
$$U = \{ ( 0,2 ) , ( 0,3 ) , ( 1,2 ) \} .$$ 
Is U transitive? Justify your answer.

Accepted Answer

@#LAT-DLM& is transitive by de

Question 13

Find a positive inverse for 7 modulo 48. (That is, ﬁnd a positive integer n such that 7n ≡ 1 (mod 48).

Accepted Answer

Step 1: @#LAT-DLM&, and so @#LAT-DLM&.
Step 2: @#LAT-DLM&, a

Question 14

Define a relation $R$ on the set $\{ 1,2,3,4 \}$ as follows:
$$R = \{ ( 1,4 ) , ( 2,3 ) , ( 2,4 ) , ( 4,1 ) , ( 2,1 ) , ( 1,2 ) , ( 3,2 ) \}$$
(a) Is $R$ symmetric? Justify your answer.
(b) Is $R$ transitive? Justify your answer.

Accepted Answer

a. @#LAT-DLM& is not symmetric

Question 15

Let A = {1, 2, 3, 4}. The following relation R is an equivalence relation on A:
R = {(1, 1), (1, 3), (1, 4), (2, 2), (3, 1), (3, 3), (3, 4), (4, 1), (4, 3), (4, 4)}.
(a) Draw the directed graph of R.
(b) Find the distinct equivalence classes of R.

Accepted Answer

The answer of Let A = {1, 2, 3, 4}....

Question 16

Prove directly from the definition of congruence modulo $n$ that if $a , c$, and $n$ are integers, $n > 1$, and $a \equiv c ( \bmod n )$, then $a ^ { 3 } \equiv c ^ { 3 } ( \bmod n )$.

Accepted Answer

Proof: Suppose @#LAT-DLM&, and @#LAT-DLM& are integers with

Question 17

Let S be the set of all strings of 0's and 1's of length 3. Deﬁne a relation R on S as follows: for all strings s and t in S,
 
(a) Prove that R is an equivalence relation on S.
(b) Find the distinct equivalence classes of R.

Accepted Answer

a. Proof:
R is reﬂexive because for each

Question 18

Define a relation $R$ from $\{ a , b , c \}$ to $\{ u , v \}$ as follows: $R = \{ ( a , v ) , ( b , u ) \}$.
(a) Draw an arrow diagram for $R$.
(b) Is $R$ a function? Why or why not?

Accepted Answer

a. @#IMG-DLM& b. R is not a function bec

Question 19

Define a relation $T$ from $\mathbf { R }$ to $\mathbf { R }$ as follows: for all $( x , y ) \in \mathbf { R } \times \mathbf { R } , x T y \Leftrightarrow y > x + 1$.
(a) Is $( 1,0 ) \in T ?$ Is $( 0,1 ) \in T ?$ Is $( - 2,5 ) \in T ?$ Is $( - 3 , - 4 ) \in T ?$
(b) Sketch the graph of $T$ in the Cartesian plane.

Accepted Answer

a. @#LAT-DLM& becaus

$\text { Use the fact that } 29 = 16 + 8 + 4 + 1 \text { to compute } 18 ^ { 29 } \bmod 65 \text {. }$

Let $A = \{ 2,3,4,5,6,7,8 \}$ and define a relation $R$ on $A$ as follows: for all $x , y \in A$ , $x R y \Leftrightarrow 3 \mid ( x - y ) .$ (a) Is $7 R 2$ ? Is $13 R 4$ ? Is $2 R 5$ ? Is $8 R 8$ ? (b) Draw the directed graph of $R$ .

Let $A = \{ 0,1,2,3 \}$ and define a relation $R$ on $A$ as follows: $R = \{ ( 0,2 ) , ( 0,3 ) , ( 2,0 ) , ( 2,1 ) \}$ . (a) Draw the directed graph of $R$ . (b) Is $R$ reflexive? Explain. (c) Is $R$ symmetric? Explain. (d) Is $R$ transitive? Explain.

Let $B = \{ 0,1,2,3 \}$ and define a relation $U$ on $B$ by $U = \{ ( 0,2 ) , ( 0,3 ) , ( 2,0 ) , ( 2,1 ) \}$ Is $U$ transitive? Justify your answer.

Define a relation $R$ from $\{ a , b , c \}$ to $\{ u , v \}$ as follows: $R = \{ ( a , u ) , ( b , u ) , ( c , v ) \}$ . (a) Draw an arrow diagram for $R$ . (b) Is $R$ a function? Why or why not?

An RSA cipher has public key pq = 65 and e = 7. (a) Translate the message YES into its numeric equivalent, and use the formula $C = M ^ { e }$ (mod pq) to encrypt the message. (b) Decrypt the ciphertext 50 16 and translate the result into letters of the alphabet to discover the message.

Define a relation $R$ from $\{ a , b , c \}$ to $\{ u , v \}$ as follows: $R = \{ ( a , v ) , ( b , u ) , ( b , v ) , ( c , u ) \}$ . (a) Draw an arrow diagram for $R$ . (b) Is $R$ a function? Why or why not?

Deﬁne a relation S on the set of positive integers as follows: for all positive integers m and n, $m S n \Leftrightarrow m \mid n$ (a) Is S reﬂexive? Justify your answer. (b) Is S symmetric? Justify your answer.

Let $A = \{ 3,4,5,6,7 \}$ and define a relation $R$ on $A$ as follows: for all $x , y \in A$ , $x R y \Leftrightarrow 2 \mid ( x - y ) .$ (a) Is 6 R 3? Is 4 R 6? (b) Draw the directed graph of R.

Define a relation $T$ on $\mathbf { R }$ as follows: for all $x$ and $y$ in $\mathbf { R } , x T y$ if and only if $x ^ { 2 } = y ^ { 2 }$ . Then $T$ is an equivalence relation on $\mathbf { R }$ . (a) Prove that T is an equivalence relation on R. (b) Find the distinct equivalence classes of T.

Let $B = \{ 0,1,2,3 \}$ and define a relation $U$ on $B$ by $U = \{ ( 0,2 ) , ( 0,3 ) , ( 1,2 ) \} .$ Is U transitive? Justify your answer.

Find a positive inverse for 7 modulo 48. (That is, ﬁnd a positive integer n such that 7n ≡ 1 (mod 48).

Define a relation $R$ on the set $\{ 1,2,3,4 \}$ as follows: $R = \{ ( 1,4 ) , ( 2,3 ) , ( 2,4 ) , ( 4,1 ) , ( 2,1 ) , ( 1,2 ) , ( 3,2 ) \}$ (a) Is $R$ symmetric? Justify your answer. (b) Is $R$ transitive? Justify your answer.

Let A = {1, 2, 3, 4}. The following relation R is an equivalence relation on A: R = {(1, 1), (1, 3), (1, 4), (2, 2), (3, 1), (3, 3), (3, 4), (4, 1), (4, 3), (4, 4)}. (a) Draw the directed graph of R. (b) Find the distinct equivalence classes of R.

Prove directly from the definition of congruence modulo $n$ that if $a , c$ , and $n$ are integers, $n > 1$ , and $a \equiv c ( \bmod n )$ , then $a ^ { 3 } \equiv c ^ { 3 } ( \bmod n )$ .

Let S be the set of all strings of 0's and 1's of length 3. Deﬁne a relation R on S as follows: for all strings s and t in S, (a) Prove that R is an equivalence relation on S. (b) Find the distinct equivalence classes of R.

Define a relation $R$ from $\{ a , b , c \}$ to $\{ u , v \}$ as follows: $R = \{ ( a , v ) , ( b , u ) \}$ . (a) Draw an arrow diagram for $R$ . (b) Is $R$ a function? Why or why not?

Speaking Mathematically

The Logic of Compound Statements

The Logic of Quantified Statements

Elementary Number Theory and Methods of Proof

Sequences, Mathematical Induction, and Recursion

Set Theory

Functions

Counting and Probability

Graphs and Trees

Analyzing Algorithm Efficiency

Regular Expressions and Finite State Automata

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Exam 8: Relations

Use the fact that 29=16+8+4+1 to compute 1829 mod 65. \text { Use the fact that } 29 = 16 + 8 + 4 + 1 \text { to compute } 18 ^ { 29 } \bmod 65 \text {. } Use the fact that 29=16+8+4+1 to compute 1829mod65.

Let B={0,1,2,3}B = \{ 0,1,2,3 \}B={0,1,2,3} and define a relation UUU on BBB by U={(0,2),(0,3),(2,0),(2,1)}U = \{ ( 0,2 ) , ( 0,3 ) , ( 2,0 ) , ( 2,1 ) \}U={(0,2),(0,3),(2,0),(2,1)} Is UUU transitive? Justify your answer.

Define a relation RRR from {a,b,c}\{ a , b , c \}{a,b,c} to {u,v}\{ u , v \}{u,v} as follows: R={(a,u),(b,u),(c,v)}R = \{ ( a , u ) , ( b , u ) , ( c , v ) \}R={(a,u),(b,u),(c,v)} . (a) Draw an arrow diagram for RRR . (b) Is RRR a function? Why or why not?

An RSA cipher has public key pq = 65 and e = 7. (a) Translate the message YES into its numeric equivalent, and use the formula C=MeC = M ^ { e }C=Me (mod pq) to encrypt the message. (b) Decrypt the ciphertext 50 16 and translate the result into letters of the alphabet to discover the message.

Define a relation RRR from {a,b,c}\{ a , b , c \}{a,b,c} to {u,v}\{ u , v \}{u,v} as follows: R={(a,v),(b,u),(b,v),(c,u)}R = \{ ( a , v ) , ( b , u ) , ( b , v ) , ( c , u ) \}R={(a,v),(b,u),(b,v),(c,u)} . (a) Draw an arrow diagram for RRR . (b) Is RRR a function? Why or why not?

Deﬁne a relation S on the set of positive integers as follows: for all positive integers m and n, mSn⇔m∣nm S n \Leftrightarrow m \mid nmSn⇔m∣n (a) Is S reﬂexive? Justify your answer. (b) Is S symmetric? Justify your answer.

Let A={3,4,5,6,7}A = \{ 3,4,5,6,7 \}A={3,4,5,6,7} and define a relation RRR on AAA as follows: for all x,y∈Ax , y \in Ax,y∈A , xRy⇔2∣(x−y).x R y \Leftrightarrow 2 \mid ( x - y ) .xRy⇔2∣(x−y). (a) Is 6 R 3? Is 4 R 6? (b) Draw the directed graph of R.

Let B={0,1,2,3}B = \{ 0,1,2,3 \}B={0,1,2,3} and define a relation UUU on BBB by U={(0,2),(0,3),(1,2)}.U = \{ ( 0,2 ) , ( 0,3 ) , ( 1,2 ) \} .U={(0,2),(0,3),(1,2)}. Is U transitive? Justify your answer.

Find a positive inverse for 7 modulo 48. (That is, ﬁnd a positive integer n such that 7n ≡ 1 (mod 48).

Let A = {1, 2, 3, 4}. The following relation R is an equivalence relation on A: R = {(1, 1), (1, 3), (1, 4), (2, 2), (3, 1), (3, 3), (3, 4), (4, 1), (4, 3), (4, 4)}. (a) Draw the directed graph of R. (b) Find the distinct equivalence classes of R.

Prove directly from the definition of congruence modulo nnn that if a,ca , ca,c , and nnn are integers, n>1n > 1n>1 , and a≡c( mod n)a \equiv c ( \bmod n )a≡c(modn) , then a3≡c3( mod n)a ^ { 3 } \equiv c ^ { 3 } ( \bmod n )a3≡c3(modn) .

Let S be the set of all strings of 0's and 1's of length 3. Deﬁne a relation R on S as follows: for all strings s and t in S, (a) Prove that R is an equivalence relation on S. (b) Find the distinct equivalence classes of R.

Define a relation RRR from {a,b,c}\{ a , b , c \}{a,b,c} to {u,v}\{ u , v \}{u,v} as follows: R={(a,v),(b,u)}R = \{ ( a , v ) , ( b , u ) \}R={(a,v),(b,u)} . (a) Draw an arrow diagram for RRR . (b) Is RRR a function? Why or why not?

Speaking Mathematically

The Logic of Compound Statements

The Logic of Quantified Statements

Elementary Number Theory and Methods of Proof

Sequences, Mathematical Induction, and Recursion

Set Theory

Functions

Counting and Probability

Graphs and Trees

Analyzing Algorithm Efficiency

Regular Expressions and Finite State Automata

Filters

$\text { Use the fact that } 29 = 16 + 8 + 4 + 1 \text { to compute } 18 ^ { 29 } \bmod 65 \text {. }$

Let $B = \{ 0,1,2,3 \}$ and define a relation $U$ on $B$ by $U = \{ ( 0,2 ) , ( 0,3 ) , ( 2,0 ) , ( 2,1 ) \}$ Is $U$ transitive? Justify your answer.

Define a relation $R$ from $\{ a , b , c \}$ to $\{ u , v \}$ as follows: $R = \{ ( a , u ) , ( b , u ) , ( c , v ) \}$ . (a) Draw an arrow diagram for $R$ . (b) Is $R$ a function? Why or why not?

An RSA cipher has public key pq = 65 and e = 7. (a) Translate the message YES into its numeric equivalent, and use the formula $C = M ^ { e }$ (mod pq) to encrypt the message. (b) Decrypt the ciphertext 50 16 and translate the result into letters of the alphabet to discover the message.

Define a relation $R$ from $\{ a , b , c \}$ to $\{ u , v \}$ as follows: $R = \{ ( a , v ) , ( b , u ) , ( b , v ) , ( c , u ) \}$ . (a) Draw an arrow diagram for $R$ . (b) Is $R$ a function? Why or why not?

Deﬁne a relation S on the set of positive integers as follows: for all positive integers m and n, $m S n \Leftrightarrow m \mid n$ (a) Is S reﬂexive? Justify your answer. (b) Is S symmetric? Justify your answer.

Let $A = \{ 3,4,5,6,7 \}$ and define a relation $R$ on $A$ as follows: for all $x , y \in A$ , $x R y \Leftrightarrow 2 \mid ( x - y ) .$ (a) Is 6 R 3? Is 4 R 6? (b) Draw the directed graph of R.

Let $B = \{ 0,1,2,3 \}$ and define a relation $U$ on $B$ by $U = \{ ( 0,2 ) , ( 0,3 ) , ( 1,2 ) \} .$ Is U transitive? Justify your answer.

Prove directly from the definition of congruence modulo $n$ that if $a , c$ , and $n$ are integers, $n > 1$ , and $a \equiv c ( \bmod n )$ , then $a ^ { 3 } \equiv c ^ { 3 } ( \bmod n )$ .

Define a relation $R$ from $\{ a , b , c \}$ to $\{ u , v \}$ as follows: $R = \{ ( a , v ) , ( b , u ) \}$ . (a) Draw an arrow diagram for $R$ . (b) Is $R$ a function? Why or why not?