Question 1

Give an example of a function $f : \mathbf { Z } \rightarrow \mathbf { N }$ that is both 1-1 and onto  $\text { N }$ .

Accepted Answer

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

Question 2

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

Accepted Answer

Not a func

Question 3

ﬁnd the inverse of the function f or else explain why the function has no inverse.
-$$f : \mathbf { R } \rightarrow \mathbf { R } \text { where } f ( x ) = \lfloor 2 x \rfloor$$

Accepted Answer

The answer of ﬁnd the inverse of the function f...

Question 4

Prove or disprove: if  A, B , and  C  are sets, then $A - ( B \cap C ) = ( A - B ) \cap ( A - C )$

Accepted Answer

A) True 
 B)False

Question 5

determine whether the rule describes a function with the given domain and codomain.
-$$f : \mathbf { N } \rightarrow \mathbf { N } \text { where } f ( n ) = \sqrt { n }$$

Accepted Answer

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

Question 6

suppose $$A = \{ a , b , c \}$$ Mark the statement TRUE or FALSE.
-$$\{ a , b \} \in A \times A$$

Accepted Answer

A) True 
 B)False

Question 7

Show that (0, 1] and R have the same cardinality.

Accepted Answer

The answer of Show that (0, 1] and R have...

Question 8

ﬁnd a recurrence relation with initial condition(s) satisﬁed by the sequence. Assume a0
is the ﬁrst term of the sequence.
-$$a _ { n } = 2 ^ { n }$$

Accepted Answer

The answer of ﬁnd a recurrence relation with initial condition(s)...

Question 9

ﬁnd a recurrence relation with initial condition(s) satisﬁed by the sequence. Assume a0
is the ﬁrst term of the sequence.
-$$a _ { n } = 2 ^ { n } + 1$$

Accepted Answer

The answer of ﬁnd a recurrence relation with initial condition(s)...

Question 10

For each of the pairs of sets in 1-3 determine whether the ﬁrst is a subset of the second, the second is a subset
of the ﬁrst, or neither is a subset of the other.
-The set of animals living in the ocean, the set of ﬁsh.

Accepted Answer

Neither is

Question 11

suppose $$A = \{ x , y \} \text { and } B = \{ x , \{ x \} \} \text {. Mark the statement TRUE or FALSE. }$$
-$$| \mathcal { P } ( A ) | = 4$$

Accepted Answer

A) True 
 B)False

Question 12

determine whether each of the following sets is countable or uncountable. For those that are countably
inﬁnite exhibit a one-to-one correspondence between the set of positive integers and that set.
-The set of irrational numbers between $$\sqrt { 2 } \text { and } \pi / 2$$

Accepted Answer

The answer of determine whether each of the following sets...

Question 13

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

Accepted Answer

A) True 
 B)False

Question 14

mark each statement TRUE or FALSE. Assume that the statement applies to all sets.
-$$\text { There is a set } A \text { such that } | \mathcal { P } ( A ) | = 12$$

Accepted Answer

A) True 
 B)False

Question 15

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . .
-1, 0.9, 0.8, 0.7, 0.6, . . . .

Accepted Answer

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

Question 16

determine whether the given set is the power set of some set. If the set is a power set, give
the set of which it is a power set.
-Prove that $$\overline { \bar { S } \cup \bar { T } } = S \cap T \text { for all sets } S \text { and } T \text {. }$$

Accepted Answer

The answer of determine whether the given set is the...

Question 17

Suppose $g : \mathbf { R } \rightarrow \mathbf { R }$ where $g ( x ) = \left\lfloor \frac { x - 1 } { 2 } \right\rfloor$
(a) If $S = \{ x \mid 1 \leq x \leq 6 \}$
, find $g ( S )$ .
(b) If $T$={2} , find $g ^ { - 1 } ( T )$

Accepted Answer

(a)(b) {[5

Question 18

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size.
-$$\mathcal { P } ( \{ a , b , c , d \} ) \text {, where } \mathcal { P } \text { denotes the power set. }$$

Accepted Answer

The answer of determine whether the set is ﬁnite or...

Question 19

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

Accepted Answer

A) True 
 B)False

Question 20

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

Accepted Answer

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

Give an example of a function $f : \mathbf { Z } \rightarrow \mathbf { N }$ that is both 1-1 and onto $\text { N }$ .

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

ﬁnd the inverse of the function f or else explain why the function has no inverse. - $f : \mathbf { R } \rightarrow \mathbf { R } \text { where } f ( x ) = \lfloor 2 x \rfloor$

Prove or disprove: if A, B , and C are sets, then $A - ( B \cap C ) = ( A - B ) \cap ( A - C )$

determine whether the rule describes a function with the given domain and codomain. - $f : \mathbf { N } \rightarrow \mathbf { N } \text { where } f ( n ) = \sqrt { n }$

suppose $A = \{ a , b , c \}$ Mark the statement TRUE or FALSE. - $\{ a , b \} \in A \times A$

Show that (0, 1] and R have the same cardinality.

ﬁnd a recurrence relation with initial condition(s) satisﬁed by the sequence. Assume a0 is the ﬁrst term of the sequence. - $a _ { n } = 2 ^ { n }$

ﬁnd a recurrence relation with initial condition(s) satisﬁed by the sequence. Assume a0 is the ﬁrst term of the sequence. - $a _ { n } = 2 ^ { n } + 1$

For each of the pairs of sets in 1-3 determine whether the ﬁrst is a subset of the second, the second is a subset of the ﬁrst, or neither is a subset of the other. -The set of animals living in the ocean, the set of ﬁsh.

suppose $A = \{ x , y \} \text { and } B = \{ x , \{ x \} \} \text {. Mark the statement TRUE or FALSE. }$ - $| \mathcal { P } ( A ) | = 4$

determine whether each of the following sets is countable or uncountable. For those that are countably inﬁnite exhibit a one-to-one correspondence between the set of positive integers and that set. -The set of irrational numbers between $\sqrt { 2 } \text { and } \pi / 2$

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

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $\text { There is a set } A \text { such that } | \mathcal { P } ( A ) | = 12$

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . . -1, 0.9, 0.8, 0.7, 0.6, . . . .

determine whether the given set is the power set of some set. If the set is a power set, give the set of which it is a power set. -Prove that $\overline { \bar { S } \cup \bar { T } } = S \cap T \text { for all sets } S \text { and } T \text {. }$

Suppose $g : \mathbf { R } \rightarrow \mathbf { R }$ where $g ( x ) = \left\lfloor \frac { x - 1 } { 2 } \right\rfloor$ (a) If $S = \{ x \mid 1 \leq x \leq 6 \}$ , find $g ( S )$ . (b) If $T$ ={2} , find $g ^ { - 1 } ( T )$

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - $\mathcal { P } ( \{ a , b , c , d \} ) \text {, where } \mathcal { P } \text { denotes the power set. }$

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

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

The Foundations: Logic and Proofs

Algorithms

Number Theory and Cryptography

Induction and Recursion

Counting

Discrete Probability

Advanced Counting Techniques

Relations

Graphs

Trees

Boolean Algebra

Modeling Computation

Filters

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

Give an example of a function f:Z→Nf : \mathbf { Z } \rightarrow \mathbf { N }f:Z→N that is both 1-1 and onto N \text { N } N .

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

ﬁnd the inverse of the function f or else explain why the function has no inverse. - f:R→R where f(x)=⌊2x⌋f : \mathbf { R } \rightarrow \mathbf { R } \text { where } f ( x ) = \lfloor 2 x \rfloorf:R→R where f(x)=⌊2x⌋

Prove or disprove: if A, B , and C are sets, then A−(B∩C)=(A−B)∩(A−C)A - ( B \cap C ) = ( A - B ) \cap ( A - C )A−(B∩C)=(A−B)∩(A−C)

determine whether the rule describes a function with the given domain and codomain. - f:N→N where f(n)=nf : \mathbf { N } \rightarrow \mathbf { N } \text { where } f ( n ) = \sqrt { n }f:N→N where f(n)=n​

suppose A={a,b,c}A = \{ a , b , c \}A={a,b,c} Mark the statement TRUE or FALSE. - {a,b}∈A×A\{ a , b \} \in A \times A{a,b}∈A×A

Show that (0, 1] and R have the same cardinality.

ﬁnd a recurrence relation with initial condition(s) satisﬁed by the sequence. Assume a0 is the ﬁrst term of the sequence. - an=2na _ { n } = 2 ^ { n }an​=2n

ﬁnd a recurrence relation with initial condition(s) satisﬁed by the sequence. Assume a0 is the ﬁrst term of the sequence. - an=2n+1a _ { n } = 2 ^ { n } + 1an​=2n+1

For each of the pairs of sets in 1-3 determine whether the ﬁrst is a subset of the second, the second is a subset of the ﬁrst, or neither is a subset of the other. -The set of animals living in the ocean, the set of ﬁsh.

suppose A={x,y} and B={x,{x}}. Mark the statement TRUE or FALSE. A = \{ x , y \} \text { and } B = \{ x , \{ x \} \} \text {. Mark the statement TRUE or FALSE. }A={x,y} and B={x,{x}}. Mark the statement TRUE or FALSE. - ∣P(A)∣=4| \mathcal { P } ( A ) | = 4∣P(A)∣=4

determine whether each of the following sets is countable or uncountable. For those that are countably inﬁnite exhibit a one-to-one correspondence between the set of positive integers and that set. -The set of irrational numbers between 2 and π/2\sqrt { 2 } \text { and } \pi / 22​ and π/2

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)\{ \varnothing \} \in \mathcal { P } ( A ){∅}∈P(A)

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - There is a set A such that ∣P(A)∣=12\text { There is a set } A \text { such that } | \mathcal { P } ( A ) | = 12 There is a set A such that ∣P(A)∣=12

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . . -1, 0.9, 0.8, 0.7, 0.6, . . . .

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - P({a,b,c,d}), where P denotes the power set. \mathcal { P } ( \{ a , b , c , d \} ) \text {, where } \mathcal { P } \text { denotes the power set. }P({a,b,c,d}), where P denotes the power set.

suppose A={a,b,c} and B={b,{c}}. Mark the statement TRUE or FALSE. A = \{ a , b , c \} \text { and } B = \{ b , \{ c \} \} \text {. Mark the statement TRUE or FALSE. }A={a,b,c} and B={b,{c}}. Mark the statement TRUE or FALSE. - {c}⊆B\{ c \} \subseteq B{c}⊆B

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

The Foundations: Logic and Proofs

Algorithms

Number Theory and Cryptography

Induction and Recursion

Counting

Discrete Probability

Advanced Counting Techniques

Relations

Graphs

Trees

Boolean Algebra

Modeling Computation

Filters

Give an example of a function $f : \mathbf { Z } \rightarrow \mathbf { N }$ that is both 1-1 and onto $\text { N }$ .

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

ﬁnd the inverse of the function f or else explain why the function has no inverse. - $f : \mathbf { R } \rightarrow \mathbf { R } \text { where } f ( x ) = \lfloor 2 x \rfloor$

Prove or disprove: if A, B , and C are sets, then $A - ( B \cap C ) = ( A - B ) \cap ( A - C )$

determine whether the rule describes a function with the given domain and codomain. - $f : \mathbf { N } \rightarrow \mathbf { N } \text { where } f ( n ) = \sqrt { n }$

suppose $A = \{ a , b , c \}$ Mark the statement TRUE or FALSE. - $\{ a , b \} \in A \times A$

ﬁnd a recurrence relation with initial condition(s) satisﬁed by the sequence. Assume a0 is the ﬁrst term of the sequence. - $a _ { n } = 2 ^ { n }$

ﬁnd a recurrence relation with initial condition(s) satisﬁed by the sequence. Assume a0 is the ﬁrst term of the sequence. - $a _ { n } = 2 ^ { n } + 1$

suppose $A = \{ x , y \} \text { and } B = \{ x , \{ x \} \} \text {. Mark the statement TRUE or FALSE. }$ - $| \mathcal { P } ( A ) | = 4$

determine whether each of the following sets is countable or uncountable. For those that are countably inﬁnite exhibit a one-to-one correspondence between the set of positive integers and that set. -The set of irrational numbers between $\sqrt { 2 } \text { and } \pi / 2$

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

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $\text { There is a set } A \text { such that } | \mathcal { P } ( A ) | = 12$

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - $\mathcal { P } ( \{ a , b , c , d \} ) \text {, where } \mathcal { P } \text { denotes the power set. }$

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

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