Question 1

Adapt the Cantor diagonalization argument to show that the set of positive real numbers less than 1 with
decimal representations consisting only of 0s and 1s is uncountable.

Accepted Answer

The answer of Adapt the Cantor diagonalization argument to show...

Question 2

Verify that $a _ { n } = 3 ^ { n }$ is a solution to the recurrence relation $a _ { n } = 4 a _ { n - 1 } - 3 a _ { n - 2 }$

Accepted Answer

The answer of Verify that \(a _ { n }...

Question 3

Find $\bigcap _ { i=1 } ^ { + \infty } ( i , \infty )$

Accepted Answer

The answer of Find \(\bigcap _ { i=1 } ^...

Question 4

suppose $$A = \{ 1,2,3,4,5 \} \text {. Mark the statement TRUE or FALSE. }$$
-$$\{ \emptyset \} \subseteq \mathcal { P } ( A )$$

Accepted Answer

A) True 
 B)False

Question 5

Suppose B=$\left( \begin{array} { l l } 
6 & 2 \
3 & 1
\end{array} \right)$  and C=$\left( \begin{array} { l l } 
2 & 1 \
0 & 6
\end{array} \right)$ .  Find a matrix  A  such that  AB=C  or prove that no such matrix exists.

Accepted Answer

The answer of Suppose B=\(\left( \begin{array} { l l }...

Question 6

Let $\mathbf { A } = \left( \begin{array} { c c } 1 & m \ 0 & 1 \end{array} \right)$Find Aⁿ where n is a positive integer.

Accepted Answer

The answer of Let \(\mathbf { A } = \left(...

Question 7

Find the sum 2 + 1/2 + 1/8 + 1/32 + 1/128 + · · · .

Accepted Answer

The answer of Find the sum 2 + 1/2 +...

Question 8

Give an example of a function $f : Z \rightarrow Z$
 that is 1-1 and not onto $\text { Z }$

Accepted Answer

The answer of Give an example of a function \(f...

Question 9

determine whether the rule describes a function with the given domain and codomain.
-$$F : \mathbf { Z } \rightarrow \mathbf { Z } \text { where } F ( x ) = \frac { 1 } { x ^ { 2 } - 5 }$$

Accepted Answer

The answer of determine whether the rule describes a function...

Question 10

use a Venn diagram to determine which relationship, $$\subseteq , = \text {, or } \supseteq \text {, }$$ is true for the pair of sets.
-$$A \cup ( B \cap C ) , ( A \cup B ) \cap C .$$

Accepted Answer

The answer of use a Venn diagram to determine which...

Question 11

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side).

Accepted Answer

The answer of Prove that \(\overline { A \cap B...

Question 12

suppose $$A = \{ 1,2,3,4,5 \} \text {. Mark the statement TRUE or FALSE. }$$
-$$\{ 1 \} \in \mathcal { P } ( A )$$

Accepted Answer

A) True 
 B)False

Question 13

mark each statement TRUE or FALSE. Assume that the statement applies to all sets.
-$$A \oplus A = A \text {. }$$

Accepted Answer

A) True 
 B)False

Question 14

suppose g: A → B and $$f : B \rightarrow C \text { where } A = \{ 1,2,3,4 \} , B = \{ a , b , c \} , C = \{ 2,8,10 \} \text {, }$$
-$$\text { Find } f \circ f ^ { - 1 } \text {. }$$

Accepted Answer

The answer of suppose g: A → B and \[f...

Question 15

Suppose $S$ ={1,2,3,4,5} . Find $| \mathcal { P } ( S ) |$ .

Accepted Answer

The answer of Suppose $S$ ={1,2,3,4,5} . Find \(| \mathcal...

Question 16

suppose $$A = \{ a , b , c \} \text { and } B = \{ b , \{ c \} \} \text {. Mark the statement TRUE or FALSE. }$$
-$$\{ \{ \{ c \} \} \} \subseteq \mathcal { P } ( B )$$

Accepted Answer

A) True 
 B)False

Question 17

Give an example of a function f: N → Z that is onto Z and not 1-1.

Accepted Answer

The answer of Give an example of a function f:...

Question 18

describe each sequence recursively. Include initial conditions and assume that the
sequences begin with a1.
-0.1, 0.11, 0.111, 0.1111, . . . .

Accepted Answer

The answer of describe each sequence recursively. Include initial conditions...

Question 19

Give an example of a function $f : \mathbf { Z } \rightarrow \mathbf { N }$ that is 1-1 and not onto  $\mathbf { N }$ .

Accepted Answer

The answer of Give an example of a function \(f...

Question 20

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . .
-3, 3, 3, 3, 3, . . . .

Accepted Answer

The answer of ﬁnd a formula that generates the following...

Exam 2: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Adapt the Cantor diagonalization argument to show that the set of positive real numbers less than 1 with decimal representations consisting only of 0s and 1s is uncountable.

Verify that $a _ { n } = 3 ^ { n }$ is a solution to the recurrence relation $a _ { n } = 4 a _ { n - 1 } - 3 a _ { n - 2 }$

Find $\bigcap _ { i=1 } ^ { + \infty } ( i , \infty )$

suppose $A = \{ 1,2,3,4,5 \} \text {. Mark the statement TRUE or FALSE. }$ - $\{ \emptyset \} \subseteq \mathcal { P } ( A )$

Suppose B= $\left( \begin{array} { l l } 6 & 2 \\3 & 1\end{array} \right)$ and C= $\left( \begin{array} { l l } 2 & 1 \\0 & 6\end{array} \right)$ . Find a matrix A such that AB=C or prove that no such matrix exists.

Let $\mathbf { A } = \left( \begin{array} { c c } 1 & m \\0 & 1\end{array} \right)$ Find Aⁿ where n is a positive integer.

Find the sum 2 + 1/2 + 1/8 + 1/32 + 1/128 + · · · .

Give an example of a function $f : Z \rightarrow Z$ that is 1-1 and not onto $\text { Z }$

determine whether the rule describes a function with the given domain and codomain. - $F : \mathbf { Z } \rightarrow \mathbf { Z } \text { where } F ( x ) = \frac { 1 } { x ^ { 2 } - 5 }$

use a Venn diagram to determine which relationship, $\subseteq , = \text {, or } \supseteq \text {, }$ is true for the pair of sets. - $A \cup ( B \cap C ) , ( A \cup B ) \cap C .$

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side).

suppose $A = \{ 1,2,3,4,5 \} \text {. Mark the statement TRUE or FALSE. }$ - $\{ 1 \} \in \mathcal { P } ( A )$

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $A \oplus A = A \text {. }$

suppose g: A → B and $f : B \rightarrow C \text { where } A = \{ 1,2,3,4 \} , B = \{ a , b , c \} , C = \{ 2,8,10 \} \text {, }$ - $\text { Find } f \circ f ^ { - 1 } \text {. }$

Suppose $S$ ={1,2,3,4,5} . Find $| \mathcal { P } ( S ) |$ .

suppose $A = \{ a , b , c \} \text { and } B = \{ b , \{ c \} \} \text {. Mark the statement TRUE or FALSE. }$ - $\{ \{ \{ c \} \} \} \subseteq \mathcal { P } ( B )$

Give an example of a function f: N → Z that is onto Z and not 1-1.

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -0.1, 0.11, 0.111, 0.1111, . . . .

Give an example of a function $f : \mathbf { Z } \rightarrow \mathbf { N }$ that is 1-1 and not onto $\mathbf { N }$ .

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . . -3, 3, 3, 3, 3, . . . .

Exam 2: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Adapt the Cantor diagonalization argument to show that the set of positive real numbers less than 1 with decimal representations consisting only of 0s and 1s is uncountable.

Verify that an=3na _ { n } = 3 ^ { n }an​=3n is a solution to the recurrence relation an=4an−1−3an−2a _ { n } = 4 a _ { n - 1 } - 3 a _ { n - 2 }an​=4an−1​−3an−2​

Find ⋂i=1+∞(i,∞)\bigcap _ { i=1 } ^ { + \infty } ( i , \infty )⋂i=1+∞​(i,∞)

suppose A={1,2,3,4,5}. Mark the statement TRUE or FALSE. A = \{ 1,2,3,4,5 \} \text {. Mark the statement TRUE or FALSE. }A={1,2,3,4,5}. Mark the statement TRUE or FALSE. - {∅}⊆P(A)\{ \emptyset \} \subseteq \mathcal { P } ( A ){∅}⊆P(A)

Suppose B= (6231)\left( \begin{array} { l l } 6 & 2 \\3 & 1\end{array} \right)(63​21​) and C= (2106)\left( \begin{array} { l l } 2 & 1 \\0 & 6\end{array} \right)(20​16​) . Find a matrix A such that AB=C or prove that no such matrix exists.

Let A=(1m01)\mathbf { A } = \left( \begin{array} { c c } 1 & m \\0 & 1\end{array} \right)A=(10​m1​) Find An where n is a positive integer.

Find the sum 2 + 1/2 + 1/8 + 1/32 + 1/128 + · · · .

Give an example of a function f:Z→Zf : Z \rightarrow Zf:Z→Z that is 1-1 and not onto Z \text { Z } Z

determine whether the rule describes a function with the given domain and codomain. - F:Z→Z where F(x)=1x2−5F : \mathbf { Z } \rightarrow \mathbf { Z } \text { where } F ( x ) = \frac { 1 } { x ^ { 2 } - 5 }F:Z→Z where F(x)=x2−51​

use a Venn diagram to determine which relationship, ⊆,=, or ⊇, \subseteq , = \text {, or } \supseteq \text {, }⊆,=, or ⊇, is true for the pair of sets. - A∪(B∩C),(A∪B)∩C.A \cup ( B \cap C ) , ( A \cup B ) \cap C .A∪(B∩C),(A∪B)∩C.

Prove that A∩B‾=Aˉ∪Bˉ\overline { A \cap B } = \bar { A } \cup \bar { B }A∩B=Aˉ∪Bˉ by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side).

suppose A={1,2,3,4,5}. Mark the statement TRUE or FALSE. A = \{ 1,2,3,4,5 \} \text {. Mark the statement TRUE or FALSE. }A={1,2,3,4,5}. Mark the statement TRUE or FALSE. - {1}∈P(A)\{ 1 \} \in \mathcal { P } ( A ){1}∈P(A)

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - A⊕A=A. A \oplus A = A \text {. }A⊕A=A.

Suppose SSS ={1,2,3,4,5} . Find ∣P(S)∣| \mathcal { P } ( S ) |∣P(S)∣ .

Give an example of a function f: N → Z that is onto Z and not 1-1.

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -0.1, 0.11, 0.111, 0.1111, . . . .

Give an example of a function f:Z→Nf : \mathbf { Z } \rightarrow \mathbf { N }f:Z→N that is 1-1 and not onto N\mathbf { N }N .

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . . -3, 3, 3, 3, 3, . . . .

Verify that $a _ { n } = 3 ^ { n }$ is a solution to the recurrence relation $a _ { n } = 4 a _ { n - 1 } - 3 a _ { n - 2 }$

Find $\bigcap _ { i=1 } ^ { + \infty } ( i , \infty )$

suppose $A = \{ 1,2,3,4,5 \} \text {. Mark the statement TRUE or FALSE. }$ - $\{ \emptyset \} \subseteq \mathcal { P } ( A )$

Suppose B= $\left( \begin{array} { l l } 6 & 2 \\3 & 1\end{array} \right)$ and C= $\left( \begin{array} { l l } 2 & 1 \\0 & 6\end{array} \right)$ . Find a matrix A such that AB=C or prove that no such matrix exists.

Let $\mathbf { A } = \left( \begin{array} { c c } 1 & m \\0 & 1\end{array} \right)$ Find Aⁿ where n is a positive integer.

Give an example of a function $f : Z \rightarrow Z$ that is 1-1 and not onto $\text { Z }$

determine whether the rule describes a function with the given domain and codomain. - $F : \mathbf { Z } \rightarrow \mathbf { Z } \text { where } F ( x ) = \frac { 1 } { x ^ { 2 } - 5 }$

use a Venn diagram to determine which relationship, $\subseteq , = \text {, or } \supseteq \text {, }$ is true for the pair of sets. - $A \cup ( B \cap C ) , ( A \cup B ) \cap C .$

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side).

suppose $A = \{ 1,2,3,4,5 \} \text {. Mark the statement TRUE or FALSE. }$ - $\{ 1 \} \in \mathcal { P } ( A )$

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $A \oplus A = A \text {. }$

Suppose $S$ ={1,2,3,4,5} . Find $| \mathcal { P } ( S ) |$ .

Give an example of a function $f : \mathbf { Z } \rightarrow \mathbf { N }$ that is 1-1 and not onto $\mathbf { N }$ .