Question 1

Is $\log _ { 2 } 5 = \log _ { 16 } 625$  Why or why not?

Accepted Answer

The two quantities are equal. Let $x = \log _ { 16 } 625$. Then, by definition of logarithm, $16 ^ { x } = 625$. Now
$$16 ^ { x } = \left( 2 ^ { 4 } \right) ^ { x } = \left( 2 ^ { x } \right) ^ { 4 } \quad \text { and } \quad 625 = 5 ^ { 4 } .$$
So
$$\left( 2 ^ { x } \right) ^ { 4 } = 5 ^ { 4 } .$$
Raising both sides to the power $1 / 4$ gives that
$$\left( \left( 2 ^ { x } \right) ^ { 4 } \right) ^ { 1 / 4 } = \left( 5 ^ { 4 } \right) ^ { 1 / 4 } , \quad \text { and thus } 2 ^ { x } = 5 .$$
Hence, by definition of logarithm,
$$\log _ { 2 } 5 = x .$$
Since both quantities equal $x$,
$$\log _ { 16 } 625 = \log _ { 2 } 5 .$$

Question 2

Let f be a function from a set X to a set Y. Deﬁne precisely (but concisely) what it means for f to be one-to-one.

Accepted Answer

Sample correct answers: A function $f$ from a set $X$ to a set $Y$ is one-to-one if, and only if, given any elements $x _ { 1 }$ and $x _ { 2 }$ in $X$, if $f \left( x _ { 1 } \right) = f \left( x _ { 2 } \right)$ then $x _ { 1 } = x _ { 2 }$.

A function $f$ from a set $X$ to a set $Y$ is one-to-one if, and only if, given any elements $x _ { 1 }$ and $x _ { 2 }$ in $X$, if $x _ { 1 } \neq x _ { 2 }$ then $f \left( x _ { 1 } \right) \neq f \left( x _ { 2 } \right)$.

A function $f$ from a set $X$ to a set $Y$ is one-to-one if, and only if, given any element $y$ in $Y$, there is at most one element $x$ in $X$ such that $f ( x ) = y$.

A function $f$ from a set $X$ to a set $Y$ is one-to-one if, and only if, each element $y$ in $Y$, is the image of at most one element $x$ in $X$.

Question 3

Let $S$ be the set of all even integers, and define a function $f : \mathbf { Z } \longrightarrow \mathbf { S }$ as follows:
$$f ( n ) = 2 n \quad \text { for all integers } n \text {. }$$
(a) Prove that $f$ is one-to-one and onto
(b) Find a formula for the inverse function $f ^ { - 1 }$.
(c) Does the set of all even integers have the same cardinality as the set of all integers? Why or why not?

Accepted Answer

a. Proof that $f$ is one-to-one: Suppose $n _ { 1 }$ and $n _ { 2 }$ are in $S$ and $f \left( n _ { 1 } \right) = f \left( n _ { 2 } \right) .$ WWe must show that $n _ { 1 } = n _ { 2 } . /$ By definition of $f , 2 n _ { 1 } = 2 n _ { 2 }$. Dividing both sides by 2 gives $n _ { 1 } = n _ { 2 }$ [as was to be shown].

Proof that $f$ is onto: Let $m$ be any even integer. [We must show that there is an integer whose image under $f$ is m.]
Let $n = \frac { m } { 2 }$. Then $n$ is an integer because $m$ is even, and
$$f ( n ) = f \left( \frac { m } { 2 } \right) = 2 \left( \frac { m } { 2 } \right) = m$$
[as was to be shown].
Conclusion: Since $f$ is both one-to-one and onto, $f$ is a one-to-one correspondence.
b. Given any even integer $m , f ^ { - 1 } ( m ) = \frac { m } { 2 }$.
c. The set of all even integers has the same cardinality as the set of all integers because there is a one-to-one correspondence from the set of all integers to the set of all even integers.

Question 4

Define a function $f : \mathbf { R } \longrightarrow \mathbf { R }$ as follows: for all real numbers $x$,
$$f ( x ) = 16 x - 5 .$$
Then $f$ is both one-to-one and onto. Find the inverse function $f ^ { - 1 }$.

Accepted Answer

@#LAT-DLM& [Scratch work: Given an eleme

Question 5

Define $F : \mathbf { R } \times \mathbf { R } \rightarrow \mathbf { R } \times \mathbf { R }$ as follows: $F ( x , y ) = ( 3 y - 1,1 - x )$ for all $( x , y )$ in $\mathbf { R } \times \mathbf { R }$.
(a) $F ( 0,0 ) = ? F ( 1,4 ) =$ ?
(b) Is $F$ one-to-one? Prove or give a counterexample.
(c) Is $F$ onto? Prove or give a counterexample.
(d) Is $F$ a one-to-one correspondence? If not, explain why not. 
If yes, find $F ^ { - 1 }$.

Accepted Answer

a. @#LAT-DLM&
b. @#LAT-DLM& is one-to-one.
Proof: Suppose @#LAT-DLM& a

Question 6

Let $S$ be the set of all nonzero real numbers. Define a function $g$ from $S$ to $S$ by the formula $g ( x ) = \frac { 1 } { x }$, for all nonzero real numbers $x$.
(a) Show that $g$ is a one-to-one correspondence from $S$ to $S$.
(b) Find $g ^ { - 1 }$.

Accepted Answer

a. Proof that g is one-to-one: Suppose x

Question 7

Let $S$ be the set of all strings in 0's and 1's, and define a function $F : S \rightarrow \mathbf { Z } ^ { \text {nonneg } }$ as follows: for all strings $s$ in $S$,
$$F ( s ) = \text { the number of } 1 \text { 's in } s .$$
(a) What is $F ( 001000 ) ? F ( 111001 )$ ? $F ( 10101 )$ ? $F ( 0100 )$ ?
(b) Is $F$ one-to-one? Prove or give a counterexample.
(c) Is $F$ onto? Prove or give a counterexample.
(d) Is $F$ a one-to-one correspondence? If so, find $F ^ { - 1 }$.

Accepted Answer

a. @#LAT-DLM&
b. @#LAT-DLM& is not one-to-one.
@#IMG-DLM& @#LAT-DLM& and @#LAT-DLM& but @#LAT-DLM&.

Question 8

(a) Draw an arrow diagram for a function that is onto but not one-to-one.
(b) Draw an arrow diagram for a function that is one-to-one but not onto.

Accepted Answer

The answer of (a) Draw an arrow diagram for a...

Question 9

$\text { Prove that if } f : X \rightarrow Y \text { and } g : Y \rightarrow Z \text { are onto, then } g \circ f : X \rightarrow Z \text { is also onto. }$

Accepted Answer

The answer to this q

Question 10

Fill in the blanks: $\log _ { 3 } \left( \frac { 1 } { 9 } \right) =$ ______because _______ .

Accepted Answer

The answer of Fill in the blanks: \(\log _ {...

Question 11

$\text { Prove that if } f : X \rightarrow Y \text { and } g : Y \rightarrow Z \text { are one-to-one, then } g \circ f : X \rightarrow Z \text { is also one-to-one. }$

Accepted Answer

The answer to this q

Question 12

Let $A = B = \{ 1,2,3 \}$, and consider the function $f : A \rightarrow B$ defined as follows: $f ( 1 ) = 3$, $f ( 2 ) = 1 , f ( 3 ) = 3$. Is $f$ onto? Why or why not?

Accepted Answer

f is not onto because there is

Question 13

Let $X = \{ 1,2,3,4,5 \}$ and $Y = \{ u , v , w , x , y \}$, and define $h : X \rightarrow Y$ as follows:
$h ( 1 ) = v , h ( 2 ) = x , h ( 3 ) = v , h ( 4 ) = v , h ( 5 ) = y .$
(a) Draw an arrow diagram for $h$.
(b) Let $A = \{ 1,2 \} , C = \{ x , v \} , D = \{ w \}$, and $E = \{ w , y \}$. Find
$$h ( A ) , h ( X ) , h ^ { - 1 } ( C ) , h ^ { - 1 } ( D ) , h ^ { - 1 } ( E ) , \text { and } h ^ { - 1 } ( Y ) .$$

Accepted Answer

The answer of Let $X = \{ 1,2,3,4,5 \}$ and...

Question 14

Prove that the set of all integers and the set of all odd integers have the same cardinality.

Accepted Answer

Proof:
Let @#LAT-DLM& denote the set of all odd in

Question 15

Let $J _ { 5 } = \{ 0,1,2,3,4 \}$ and define a function $g : J _ { 5 } \times J _ { 5 } \rightarrow J _ { 5 } \times J _ { 5 }$ as follows: For all $( a , b ) \in$ $J _ { 5 } \times J _ { 5 }$
$g ( a , b ) = ( ( 5 a - 3 ) \bmod 5 , ( 4 b + 2 ) \bmod 5 ) .$ Find $g ( 3,4 )$.

Accepted Answer

The answer of Let \(J _ { 5 } =...

Question 16

Is the set of all squares of positive integers countable? That is, is the set $S = \left\{ m \in \mathbf { Z } \mid m = k ^ { 2 } \right.$ for some positive integer $k \}$ a countable set. Justify your answer.

Accepted Answer

Most of the answer to this question is c

Question 17

Let $S$ be the set of all strings in 0's and 1's, and define a function $g$ : $S \longrightarrow \mathbf { Z }$ as follows: for each string $s$ in $S$,
$g ( s ) =$ the number of 1's in $s$ minus the number of 0 's in $s .$
(a) What is $g ( 101011 ) ? g ( 00100 )$ ?
(b) Is $g$ one-to-one? Prove or give a counterexample.
(c) Is $g$ onto? Prove or give a counterexample.

Accepted Answer

a. @#LAT-DLM&
b. @#LAT-DLM& is not one-to-one.
@#IMG-DLM&@#LAT-DLM& but @#LAT-DLM&.
[This

Question 18

Let $S$ be the set of all strings in 0 's and 1's, and define a function $g : S \longrightarrow \mathbf { Z }$ as follows: for each string $s$ in $S$,
$$g ( s ) = \text { the number of } 0 \text { 's in } s .$$
(a) What is $g ( 101011 )$ ? $g ( 00100 )$ ?
(b) Is $g$ one-to-one? Prove or give a counterexample.
(c) Is $g$ onto? Prove or give a counterexample.

Accepted Answer

a. @#LAT-DLM&
b. @#LAT-DLM& is not one-to-one.
@#IMG-DLM& @#LAT-DLM& and @#LAT-DLM& but @#LAT-DLM&.

Question 19

Let f be a function from a set X to a set Y. Deﬁne precisely (but concisely) what it means for f to be onto.

Accepted Answer

Sample correct answers:
A function f fro

Question 20

Let $X = \{ a , b , c \}$ and $Y = \{ u , v \}$. Which of the following arrow diagrams define functions from $X$ to $Y$ ?
a.
 
b.
 
c.

Accepted Answer

The only arrow diagram that deﬁnes a fun

Is $\log _ { 2 } 5 = \log _ { 16 } 625$ Why or why not?

Let f be a function from a set X to a set Y. Deﬁne precisely (but concisely) what it means for f to be one-to-one.

Define a function $f : \mathbf { R } \longrightarrow \mathbf { R }$ as follows: for all real numbers $x$ , $f ( x ) = 16 x - 5 .$ Then $f$ is both one-to-one and onto. Find the inverse function $f ^ { - 1 }$ .

Let $S$ be the set of all nonzero real numbers. Define a function $g$ from $S$ to $S$ by the formula $g ( x ) = \frac { 1 } { x }$ , for all nonzero real numbers $x$ . (a) Show that $g$ is a one-to-one correspondence from $S$ to $S$ . (b) Find $g ^ { - 1 }$ .

(a) Draw an arrow diagram for a function that is onto but not one-to-one. (b) Draw an arrow diagram for a function that is one-to-one but not onto.

$\text { Prove that if } f : X \rightarrow Y \text { and } g : Y \rightarrow Z \text { are onto, then } g \circ f : X \rightarrow Z \text { is also onto. }$

Fill in the blanks: $\log _ { 3 } \left( \frac { 1 } { 9 } \right) =$ because _ .

$\text { Prove that if } f : X \rightarrow Y \text { and } g : Y \rightarrow Z \text { are one-to-one, then } g \circ f : X \rightarrow Z \text { is also one-to-one. }$

Let $A = B = \{ 1,2,3 \}$ , and consider the function $f : A \rightarrow B$ defined as follows: $f ( 1 ) = 3$ , $f ( 2 ) = 1 , f ( 3 ) = 3$ . Is $f$ onto? Why or why not?

Prove that the set of all integers and the set of all odd integers have the same cardinality.

Let $J _ { 5 } = \{ 0,1,2,3,4 \}$ and define a function $g : J _ { 5 } \times J _ { 5 } \rightarrow J _ { 5 } \times J _ { 5 }$ as follows: For all $( a , b ) \in$ $J _ { 5 } \times J _ { 5 }$ $g ( a , b ) = ( ( 5 a - 3 ) \bmod 5 , ( 4 b + 2 ) \bmod 5 ) .$ Find $g ( 3,4 )$ .

Is the set of all squares of positive integers countable? That is, is the set $S = \left\{ m \in \mathbf { Z } \mid m = k ^ { 2 } \right.$ for some positive integer $k \}$ a countable set. Justify your answer.

Let f be a function from a set X to a set Y. Deﬁne precisely (but concisely) what it means for f to be onto.

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

Relations

Counting and Probability

Graphs and Trees

Analyzing Algorithm Efficiency

Regular Expressions and Finite State Automata

Filters

Exam 7: Functions

Is log⁡25=log⁡16625\log _ { 2 } 5 = \log _ { 16 } 625log2​5=log16​625 Why or why not?

Let f be a function from a set X to a set Y. Deﬁne precisely (but concisely) what it means for f to be one-to-one.

Define a function f:R⟶Rf : \mathbf { R } \longrightarrow \mathbf { R }f:R⟶R as follows: for all real numbers xxx , f(x)=16x−5.f ( x ) = 16 x - 5 .f(x)=16x−5. Then fff is both one-to-one and onto. Find the inverse function f−1f ^ { - 1 }f−1 .

Let SSS be the set of all nonzero real numbers. Define a function ggg from SSS to SSS by the formula g(x)=1xg ( x ) = \frac { 1 } { x }g(x)=x1​ , for all nonzero real numbers xxx . (a) Show that ggg is a one-to-one correspondence from SSS to SSS . (b) Find g−1g ^ { - 1 }g−1 .

(a) Draw an arrow diagram for a function that is onto but not one-to-one. (b) Draw an arrow diagram for a function that is one-to-one but not onto.

Fill in the blanks: log⁡3(19)=\log _ { 3 } \left( \frac { 1 } { 9 } \right) =log3​(91​)= ______because _______ .

Let A=B={1,2,3}A = B = \{ 1,2,3 \}A=B={1,2,3} , and consider the function f:A→Bf : A \rightarrow Bf:A→B defined as follows: f(1)=3f ( 1 ) = 3f(1)=3 , f(2)=1,f(3)=3f ( 2 ) = 1 , f ( 3 ) = 3f(2)=1,f(3)=3 . Is fff onto? Why or why not?

Prove that the set of all integers and the set of all odd integers have the same cardinality.

Is the set of all squares of positive integers countable? That is, is the set S={m∈Z∣m=k2S = \left\{ m \in \mathbf { Z } \mid m = k ^ { 2 } \right.S={m∈Z∣m=k2 for some positive integer k}k \}k} a countable set. Justify your answer.

Let f be a function from a set X to a set Y. Deﬁne precisely (but concisely) what it means for f to be onto.

Let X={a,b,c}X = \{ a , b , c \}X={a,b,c} and Y={u,v}Y = \{ u , v \}Y={u,v} . Which of the following arrow diagrams define functions from XXX to YYY ? a. b. c.

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

Relations

Counting and Probability

Graphs and Trees

Analyzing Algorithm Efficiency

Regular Expressions and Finite State Automata

Filters

Is $\log _ { 2 } 5 = \log _ { 16 } 625$ Why or why not?

Define a function $f : \mathbf { R } \longrightarrow \mathbf { R }$ as follows: for all real numbers $x$ , $f ( x ) = 16 x - 5 .$ Then $f$ is both one-to-one and onto. Find the inverse function $f ^ { - 1 }$ .

Let $S$ be the set of all nonzero real numbers. Define a function $g$ from $S$ to $S$ by the formula $g ( x ) = \frac { 1 } { x }$ , for all nonzero real numbers $x$ . (a) Show that $g$ is a one-to-one correspondence from $S$ to $S$ . (b) Find $g ^ { - 1 }$ .

Fill in the blanks: $\log _ { 3 } \left( \frac { 1 } { 9 } \right) =$ because _ .

Let $A = B = \{ 1,2,3 \}$ , and consider the function $f : A \rightarrow B$ defined as follows: $f ( 1 ) = 3$ , $f ( 2 ) = 1 , f ( 3 ) = 3$ . Is $f$ onto? Why or why not?

Is the set of all squares of positive integers countable? That is, is the set $S = \left\{ m \in \mathbf { Z } \mid m = k ^ { 2 } \right.$ for some positive integer $k \}$ a countable set. Justify your answer.