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

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $\overline { A \cup \bar { B } } \cup \bar { A } = \bar { A }$

(True/False)

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Question 81

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. - $\{ \varnothing , \{ a \} , \{ \varnothing \} , \{ a , \emptyset \} \}$

(Short Answer)

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Question 82

Find $\bigcup _ { i = 1 } ^ { + \infty } [ - 1 / i , 1 / i ]$

(Short Answer)

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Question 83

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

(True/False)

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Question 84

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. - $\{ \varnothing , \{ a \} \}$

(Short Answer)

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Question 85

Find the sum 1 - 1/2 + 1/4 - 1/8 + 1/16 - ... .

(Short Answer)

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Question 86

suppose g $\text { : } A \rightarrow B \text { and } f : B \rightarrow C \text { where } A = \{ a , b , c , d \} , B = \{ 1,2,3 \} , C = \{ 2,3,6,8 \}$ $\text { and } g \text { and } f \text { are defined by } g = \{ ( a , 2 ) , ( b , 1 ) , ( c , 3 ) , ( d , 2 ) \} \text { and } f = \{ ( 1,8 ) , ( 2,3 ) , ( 3,2 ) \} \text {. }$ - $\text { Find } f \circ g \text {. }$

(Short Answer)

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Question 87

Find the sum 112 + 113 + 114 + · · · + 673.

(Short Answer)

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Question 88

Rewrite $\sum _ { i = -3 } ^ { 4 } \left( i ^ { 2 } + 1 \right)$ so that the index of summation has lower limit 0 and upper limit 7 .

(Short Answer)

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Question 89

Determine whether $f$ is a function from the set of all bit strings to the set of integers if $f ( S )$ is the number of 0 bits in S .

(Short Answer)

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Question 90

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a proof using logical equivalence.

(Essay)

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Question 91

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

(True/False)

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Question 92

In questions determine whether the statement is true or false. - $\text { If } \mathbf { A } = \left( \begin{array} { l l } 3 & 5 \\1 & 2\end{array} \right) , \text { then } \mathbf { A } ^ { - 1 } = \left( \begin{array} { c c } 2 & 5 \\1 & - 3\end{array} \right)$

(True/False)

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Question 93

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

(Short Answer)

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Question 94

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -3, 2, 1, 0, −1, −2, . . . .

(Short Answer)

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Question 95

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - $\left\{ x \mid x \in \mathbf { Z } \text { and } x ^ { 2 } < 8 \right\}$

(Short Answer)

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Question 96

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

(Short Answer)

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Question 97

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $( A - C ) - ( B - C ) = A - B$

(True/False)

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(18)

Question 98

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. - $\{ \varnothing , \{ a , \varnothing \} \} .$

(Short Answer)

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(32)

Question 99

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

(True/False)

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Question 100

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $\overline { A \cup \bar { B } } \cup \bar { A } = \bar { A }$

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. - $\{ \varnothing , \{ a \} , \{ \varnothing \} , \{ a , \emptyset \} \}$

Find $\bigcup _ { i = 1 } ^ { + \infty } [ - 1 / i , 1 / i ]$

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

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. - $\{ \varnothing , \{ a \} \}$

Find the sum 1 - 1/2 + 1/4 - 1/8 + 1/16 - ... .

Find the sum 112 + 113 + 114 + · · · + 673.

Rewrite $\sum _ { i = -3 } ^ { 4 } \left( i ^ { 2 } + 1 \right)$ so that the index of summation has lower limit 0 and upper limit 7 .

Determine whether $f$ is a function from the set of all bit strings to the set of integers if $f ( S )$ is the number of 0 bits in S .

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a proof using logical equivalence.

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

In questions determine whether the statement is true or false. - $\text { If } \mathbf { A } = \left( \begin{array} { l l } 3 & 5 \\1 & 2\end{array} \right) , \text { then } \mathbf { A } ^ { - 1 } = \left( \begin{array} { c c } 2 & 5 \\1 & - 3\end{array} \right)$

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

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -3, 2, 1, 0, −1, −2, . . . .

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - $\left\{ x \mid x \in \mathbf { Z } \text { and } x ^ { 2 } < 8 \right\}$

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

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $( A - C ) - ( B - C ) = A - B$

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. - $\{ \varnothing , \{ a , \varnothing \} \} .$

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

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

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - A∪Bˉ‾∪Aˉ=Aˉ\overline { A \cup \bar { B } } \cup \bar { A } = \bar { A }A∪Bˉ∪Aˉ=Aˉ

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. - {∅,{a},{∅},{a,∅}}\{ \varnothing , \{ a \} , \{ \varnothing \} , \{ a , \emptyset \} \}{∅,{a},{∅},{a,∅}}

Find ⋃i=1+∞[−1/i,1/i]\bigcup _ { i = 1 } ^ { + \infty } [ - 1 / i , 1 / i ]⋃i=1+∞​[−1/i,1/i]

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. - {x}⊆A−B\{ x \} \subseteq A - B{x}⊆A−B

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. - {∅,{a}}\{ \varnothing , \{ a \} \}{∅,{a}}

Find the sum 1 - 1/2 + 1/4 - 1/8 + 1/16 - ... .

Find the sum 112 + 113 + 114 + · · · + 673.

Rewrite ∑i=−34(i2+1)\sum _ { i = -3 } ^ { 4 } \left( i ^ { 2 } + 1 \right)∑i=−34​(i2+1) so that the index of summation has lower limit 0 and upper limit 7 .

Determine whether fff is a function from the set of all bit strings to the set of integers if f(S)f ( S )f(S) is the number of 0 bits in S .

Prove that A∩B‾=Aˉ∪Bˉ\overline { A \cap B } = \bar { A } \cup \bar { B }A∩B=Aˉ∪Bˉ by giving a proof using logical equivalence.

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. - ∅⊆A×A.\emptyset \subseteq A \times A .∅⊆A×A.

Give an example of a function f⋅N→Zf \cdot \mathbf { N } \rightarrow \mathbf { Z }f⋅N→Z that is both 1-1 and onto ZZZ

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -3, 2, 1, 0, −1, −2, . . . .

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - {x∣x∈Z and x2<8}\left\{ x \mid x \in \mathbf { Z } \text { and } x ^ { 2 } < 8 \right\}{x∣x∈Z and x2<8}

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

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - (A−C)−(B−C)=A−B( A - C ) - ( B - C ) = A - B(A−C)−(B−C)=A−B

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. - {∅,{a,∅}}.\{ \varnothing , \{ a , \varnothing \} \} .{∅,{a,∅}}.

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

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $\overline { A \cup \bar { B } } \cup \bar { A } = \bar { A }$

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. - $\{ \varnothing , \{ a \} , \{ \varnothing \} , \{ a , \emptyset \} \}$

Find $\bigcup _ { i = 1 } ^ { + \infty } [ - 1 / i , 1 / i ]$

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

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. - $\{ \varnothing , \{ a \} \}$

Rewrite $\sum _ { i = -3 } ^ { 4 } \left( i ^ { 2 } + 1 \right)$ so that the index of summation has lower limit 0 and upper limit 7 .

Determine whether $f$ is a function from the set of all bit strings to the set of integers if $f ( S )$ is the number of 0 bits in S .

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a proof using logical equivalence.

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

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

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - $\left\{ x \mid x \in \mathbf { Z } \text { and } x ^ { 2 } < 8 \right\}$

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

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $( A - C ) - ( B - C ) = A - B$

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. - $\{ \varnothing , \{ a , \varnothing \} \} .$

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