Question 1

$$\text { Let } A = \{ a , b , c \} \text { and } B = \{ u , v \} \text {. Write } a . A \times B \text { and } b . B \times A \text {. }$$

Accepted Answer

a. $\{ ( a , u ) , ( a , v ) , ( b , u ) , ( b , v ) , ( c , u ) , ( c , v ) \}$
b. $\{ ( u , a ) , ( v , a ) , ( u , b ) , ( v , b ) , ( u , c ) , ( v , c ) \}$

Question 2

Fill in the blanks to rewrite the following statement with variables:
-Given any positive real number, there is a positive real number that is smaller.
(a) Given any positive real number r, there is ________ s such that s is _______ .
(b) For any ________ , _________ such that s < r.

Accepted Answer

a. a positive real number; smaller than r 
b. positive real number r; there is a positive real number s

Question 3

Define functions $F$ and $G$ from $\mathbf { R }$ to $\mathbf { R }$ by the following formulas:
$$F ( x ) = ( x + 1 ) ( x - 3 ) \quad \text { and } \quad G ( x ) = ( x - 2 ) ^ { 2 } - 7 .$$
Does $F = G$ ? Explain.

Accepted Answer

$F \neq G$. Note that for every real number $x$,
$G ( x ) = ( x - 2 ) ^ { 2 } - 7 = x ^ { 2 } - 4 x + 4 - 7 = x ^ { 2 } - 4 x - 3$
whereas
$F ( x ) = ( x + 1 ) ( x - 3 ) = x ^ { 2 } - 2 x - 3$.  Thus, for instance,
 $F ( 1 ) = ( 1 + 1 ) ( 1 - 3 ) = - 4 \quad \text { whereas } \quad G ( 1 ) = ( 1 - 2 ) ^ { 2 } - 7 = - 6$

Question 4

Fill in the blanks to rewrite the following statement with variables: 
-Is there an integer with a remainder of 1 when it is divided by 4 and a remainder of 3 when it is divided by 7?
(a) Is there an integer n such that n has ________ ?
(b) Does there exist _______ such that if n is divided by 4 the remainder is 1 and if ________ ?

Accepted Answer

a. a remainder of 1 when it is

Question 5

Fill in the blanks to rewrite the following statement:
-Every real number has an additive inverse.
(a) All real numbers _______.
(b) For any real number x, there is _______ for x.
(c) For all real numbers x, there is real number y such that ________ .

Accepted Answer

a. have additive inv

Question 6

Define a relation $R$ from $\mathbf { R }$ to $\mathbf { R }$ as follows: For all $( x , y ) \in \mathbf { R } \times \mathbf { R } , ( x , y ) \in R$ if, and only if, $x = y ^ { 2 } + 1$.
(a) Is $( 2,5 ) \in R$ ? Is $( 5,2 ) \in R$ ? Is (-3) $R$ 10? Is $10 R ( - 3 )$ ?
(b) Draw the graph of $R$ in the Cartesian plane.
(c) Is $R$ a function from $\mathbf { R }$ to $\mathbf { R }$ ? Explain.

Accepted Answer

a. No; Yes; No; Yes
b. Draw the graph of

Question 7

Let $A = \{ 3,5,7 \}$ and $B = \{ 15,16,17,18 \}$, and define a relation $R$ from $A$ to $B$ as follows: For all $( x , y ) \in A \times B$, 
$( x , y ) \in R \quad \Leftrightarrow \quad  \frac { y } { x }$  is an integer.
(a) Is $3 R 15$ ? Is $3 R 16 ? \quad$ Is $( 7,17 ) \in R$ ? Is $( 3,18 ) \in R$ ?
(b) Write $R$ as a set of ordered pairs.
(c) Write the domain and co-domain of $R$.
(d) Draw an arrow diagram for $R$.
(e) Is $R$ a function from $A$ to $B$ ? Explain.

Accepted Answer

a. Yes; No; No; Yes
b. @#LAT-DLM&
c. domain is @#LAT-DLM&; c

Question 8

Rewrite the following statement less formally, without using variables:
-There is an integer n such that 1/n is also an integer.

Accepted Answer

There is an integer

Question 9

(a) Is $\{ 5 \} \in \{ 1,3,5 \}$ ?
(b) Is $\{ 5 \} \subseteq \{ 1,3,5 \}$ ?
(c) Is $\{ 5 \} \in \{ \{ 1 \} , \{ 3 \} , \{ 5 \} \}$ ?
(d) Is $\{ 5 \} \subseteq \{ \{ 1 \} , \{ 3 \} , \{ 5 \} \}$ ?

Accepted Answer

(a) No
(b)

Question 10

Let $A = \{ 1,2,3,4 \}$ and $B = \{ a , b , c \}$. Define a function $G : A \rightarrow B$ as follows:
$$G = \{ ( 1 , b ) , ( 2 , c ) , ( 3 , b ) , ( 4 , c ) \}$$
(a) Find $G ( 2 )$.
(b) Draw an arrow diagram for $G$.

Accepted Answer

a. @#LAT-DLM&
b. Dra

Question 11

(a) Write in words how to read the following out loud  $\{ n \in \mathbf { Z } \mid n \text { is a factor of } 9 \}$ 
(b) Use the set-roster notation to indicate the elements in the set.

Accepted Answer

(a) The set of all integers n such that

Question 12

Fill in the blanks to rewrite the following statement:
-For all objects T, if T is a triangle then T has three sides.
(a) All triangles ________ .
(b) Every triangle _______ .
(c) If an object is a triangle, then it _______ .
(d) If T ________ , then T ________ .
(e) For all triangles T, _________ .

Accepted Answer

a. have three sides
b. has thr

Question 13

Fill in the blanks to rewrite the following statement:
-There is a positive integer that is less than or equal to every positive integer.
(a) There is a positive integer m such that m is _______ .
(b) There is a ________ such that _______ every positive integer.
(c) There is a positive integer m which satisﬁes the property that given any positive integer
n, m is ________ .

Accepted Answer

a. less than or equal to every

$\text { Let } A = \{ a , b , c \} \text { and } B = \{ u , v \} \text {. Write } a . A \times B \text { and } b . B \times A \text {. }$

Fill in the blanks to rewrite the following statement with variables: -Given any positive real number, there is a positive real number that is smaller. (a) Given any positive real number r, there is ____ s such that s is _ . (b) For any , ___ such that s < r.

Define functions $F$ and $G$ from $\mathbf { R }$ to $\mathbf { R }$ by the following formulas: $F ( x ) = ( x + 1 ) ( x - 3 ) \quad \text { and } \quad G ( x ) = ( x - 2 ) ^ { 2 } - 7 .$ Does $F = G$ ? Explain.

Fill in the blanks to rewrite the following statement: -Every real number has an additive inverse. (a) All real numbers ___. (b) For any real number x, there is _ for x. (c) For all real numbers x, there is real number y such that ______ .

Rewrite the following statement less formally, without using variables: -There is an integer n such that 1/n is also an integer.

(a) Is $\{ 5 \} \in \{ 1,3,5 \}$ ? (b) Is $\{ 5 \} \subseteq \{ 1,3,5 \}$ ? (c) Is $\{ 5 \} \in \{ \{ 1 \} , \{ 3 \} , \{ 5 \} \}$ ? (d) Is $\{ 5 \} \subseteq \{ \{ 1 \} , \{ 3 \} , \{ 5 \} \}$ ?

Let $A = \{ 1,2,3,4 \}$ and $B = \{ a , b , c \}$ . Define a function $G : A \rightarrow B$ as follows: $G = \{ ( 1 , b ) , ( 2 , c ) , ( 3 , b ) , ( 4 , c ) \}$ (a) Find $G ( 2 )$ . (b) Draw an arrow diagram for $G$ .

(a) Write in words how to read the following out loud $\{ n \in \mathbf { Z } \mid n \text { is a factor of } 9 \}$ (b) Use the set-roster notation to indicate the elements in the set.

Fill in the blanks to rewrite the following statement: -For all objects T, if T is a triangle then T has three sides. (a) All triangles ____ . (b) Every triangle _ . (c) If an object is a triangle, then it _ . (d) If T , then T . (e) For all triangles T, _____ .

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

Relations

Counting and Probability

Graphs and Trees

Analyzing Algorithm Efficiency

Regular Expressions and Finite State Automata

Filters

Exam 1: Speaking Mathematically

Let A={a,b,c} and B={u,v}. Write a.A×B and b.B×A. \text { Let } A = \{ a , b , c \} \text { and } B = \{ u , v \} \text {. Write } a . A \times B \text { and } b . B \times A \text {. } Let A={a,b,c} and B={u,v}. Write a.A×B and b.B×A.

Fill in the blanks to rewrite the following statement with variables: -Given any positive real number, there is a positive real number that is smaller. (a) Given any positive real number r, there is ________ s such that s is _______ . (b) For any ________ , _________ such that s < r.

Define functions FFF and GGG from R\mathbf { R }R to R\mathbf { R }R by the following formulas: F(x)=(x+1)(x−3) and G(x)=(x−2)2−7.F ( x ) = ( x + 1 ) ( x - 3 ) \quad \text { and } \quad G ( x ) = ( x - 2 ) ^ { 2 } - 7 .F(x)=(x+1)(x−3) and G(x)=(x−2)2−7. Does F=GF = GF=G ? Explain.

Fill in the blanks to rewrite the following statement: -Every real number has an additive inverse. (a) All real numbers _______. (b) For any real number x, there is _______ for x. (c) For all real numbers x, there is real number y such that ________ .