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

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $\text { If } A \cup C = B \cup C \text {, then } A = B \text {. }$

(True/False)

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Question 141

ﬁnd the inverse of the function f or else explain why the function has no inverse. - $f : \mathbf { Z } \rightarrow \mathbf { Z } \text { where } f ( x ) = \left\{ \begin{array} { l l } x - 2 & \text { if } x \geq 5 \\x + 1 & \text { if } x \leq 4\end{array} \right.$

(Short Answer)

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Question 142

Find $\bigcap _ { i = 1 } ^ { + \infty } \left[ 1 - \frac { 1 } { i } , 1 \right]$

(Short Answer)

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Question 143

Suppose $f : \mathbf { R } \rightarrow \mathbf { Z }$ where $f ( x ) = \lceil 2 x - 1 \rceil$ (a) Draw the graph of $f$ (b) Is $f$ 1-1 ? (Explain) (c) Is $f$ onto Z ? (Explain)

(Short Answer)

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Question 144

Find the sum 2-4+8-16+32-...-2²⁸ .

(Short Answer)

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Question 145

Find three subsets of {1,2,3,4,5,6,7,8,9 } such that the intersection of any two has size 2 and the intersection of all three has size 1 .

(Short Answer)

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Question 146

Show that $\lceil x \rceil = - \lfloor - x \rfloor$

(Essay)

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Question 147

Verify that $a _ { n } = 7 \cdot 3 ^ { n } - \pi$ is a solution to the recurrence relation $a _ { n } = 4 a _ { n - 1 } - 3 a _ { n - 2 }$

(Short Answer)

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Question 148

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. - $a _ { n } = \text { the number of subsets of a set of size } n$

(Short Answer)

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Question 149

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

(Short Answer)

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Question 150

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - $\left\{ x \mid x \in \mathbf { N } \text { and } 9 x ^ { 2 } - 1 = 0 \right\}$

(Short Answer)

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Question 151

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

(True/False)

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Question 152

Find the sum 2+4+8+16+32+...+2²⁸ .

(Short Answer)

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Question 153

In questions determine whether the statement is true or false. - $\text { If } \mathbf { A B } = \mathbf { A C } \text {, then } \mathbf { B } = \mathbf { C } \text {. }$

(True/False)

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Question 154

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

(Short Answer)

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Question 155

Suppose f: R → R and g: R → R where g(x) = 2x + 1 and g ◦ f(x) = 2x + 11. Find the rule for f.

(Short Answer)

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Question 156

Prove or disprove: For all positive real numbers $x$ and $y$ , $\lfloor x \cdot y \rfloor \leq \lfloor x \rfloor \cdot \lfloor y \rfloor$

(Essay)

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Question 157

Suppose A is a 2 X 2 matrix with real number entries such that AB=BA for all 2 X 2 matrices. What relationships must exist among the entries of A ?

(Short Answer)

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Question 158

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

(True/False)

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Question 159

Suppose B = $\left( \begin{array} { l l } 3 & 5 \\2 & 4\end{array} \right)$ 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.

(Short Answer)

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Question 160

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $\text { If } A \cup C = B \cup C \text {, then } A = B \text {. }$

ﬁnd the inverse of the function f or else explain why the function has no inverse. - $f : \mathbf { Z } \rightarrow \mathbf { Z } \text { where } f ( x ) = \left\{ \begin{array} { l l } x - 2 & \text { if } x \geq 5 \\x + 1 & \text { if } x \leq 4\end{array} \right.$

Find $\bigcap _ { i = 1 } ^ { + \infty } \left[ 1 - \frac { 1 } { i } , 1 \right]$

Suppose $f : \mathbf { R } \rightarrow \mathbf { Z }$ where $f ( x ) = \lceil 2 x - 1 \rceil$ (a) Draw the graph of $f$ (b) Is $f$ 1-1 ? (Explain) (c) Is $f$ onto Z ? (Explain)

Find the sum 2-4+8-16+32-...-2²⁸ .

Find three subsets of {1,2,3,4,5,6,7,8,9 } such that the intersection of any two has size 2 and the intersection of all three has size 1 .

Show that $\lceil x \rceil = - \lfloor - x \rfloor$

Verify that $a _ { n } = 7 \cdot 3 ^ { n } - \pi$ is a solution to the recurrence relation $a _ { n } = 4 a _ { n - 1 } - 3 a _ { n - 2 }$

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. - $a _ { n } = \text { the number of subsets of a set of size } n$

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

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - $\left\{ x \mid x \in \mathbf { N } \text { and } 9 x ^ { 2 } - 1 = 0 \right\}$

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

Find the sum 2+4+8+16+32+...+2²⁸ .

In questions determine whether the statement is true or false. - $\text { If } \mathbf { A B } = \mathbf { A C } \text {, then } \mathbf { B } = \mathbf { C } \text {. }$

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

Suppose f: R → R and g: R → R where g(x) = 2x + 1 and g ◦ f(x) = 2x + 11. Find the rule for f.

Prove or disprove: For all positive real numbers $x$ and $y$ , $\lfloor x \cdot y \rfloor \leq \lfloor x \rfloor \cdot \lfloor y \rfloor$

Suppose A is a 2 X 2 matrix with real number entries such that AB=BA for all 2 X 2 matrices. What relationships must exist among the entries of A ?

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

Suppose B = $\left( \begin{array} { l l } 3 & 5 \\2 & 4\end{array} \right)$ 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.

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. - If A∪C=B∪C, then A=B. \text { If } A \cup C = B \cup C \text {, then } A = B \text {. } If A∪C=B∪C, then A=B.

Find ⋂i=1+∞[1−1i,1]\bigcap _ { i = 1 } ^ { + \infty } \left[ 1 - \frac { 1 } { i } , 1 \right]⋂i=1+∞​[1−i1​,1]

Suppose f:R→Zf : \mathbf { R } \rightarrow \mathbf { Z }f:R→Z where f(x)=⌈2x−1⌉f ( x ) = \lceil 2 x - 1 \rceilf(x)=⌈2x−1⌉ (a) Draw the graph of fff (b) Is fff 1-1 ? (Explain) (c) Is fff onto Z ? (Explain)

Find the sum 2-4+8-16+32-...-228 .

Find three subsets of {1,2,3,4,5,6,7,8,9 } such that the intersection of any two has size 2 and the intersection of all three has size 1 .

Show that ⌈x⌉=−⌊−x⌋\lceil x \rceil = - \lfloor - x \rfloor⌈x⌉=−⌊−x⌋

Verify that an=7⋅3n−πa _ { n } = 7 \cdot 3 ^ { n } - \pian​=7⋅3n−π 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​

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. - an= the number of subsets of a set of size na _ { n } = \text { the number of subsets of a set of size } nan​= the number of subsets of a set of size n

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∘g. \text { Find } f \circ g \text {. } Find f∘g.

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - {x∣x∈N and 9x2−1=0}\left\{ x \mid x \in \mathbf { N } \text { and } 9 x ^ { 2 } - 1 = 0 \right\}{x∣x∈N and 9x2−1=0}

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - A∩(B∪C)=(A∪B)∩(A∪C)A \cap ( B \cup C ) = ( A \cup B ) \cap ( A \cup C )A∩(B∪C)=(A∪B)∩(A∪C)

Find the sum 2+4+8+16+32+...+228 .

In questions determine whether the statement is true or false. - If AB=AC, then B=C. \text { If } \mathbf { A B } = \mathbf { A C } \text {, then } \mathbf { B } = \mathbf { C } \text {. } If AB=AC, then B=C.

Find ⋂i=1+∞(1−1i,1)\bigcap _ { i = 1 } ^ { + \infty } \left( 1 - \frac { 1 } { i } , 1 \right)⋂i=1+∞​(1−i1​,1)

Suppose f: R → R and g: R → R where g(x) = 2x + 1 and g ◦ f(x) = 2x + 11. Find the rule for f.

Prove or disprove: For all positive real numbers xxx and yyy , ⌊x⋅y⌋≤⌊x⌋⋅⌊y⌋\lfloor x \cdot y \rfloor \leq \lfloor x \rfloor \cdot \lfloor y \rfloor⌊x⋅y⌋≤⌊x⌋⋅⌊y⌋

Suppose A is a 2 X 2 matrix with real number entries such that AB=BA for all 2 X 2 matrices. What relationships must exist among the entries of A ?

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. - B⊆AB \subseteq AB⊆A

Suppose B = (3524)\left( \begin{array} { l l } 3 & 5 \\2 & 4\end{array} \right)(32​54​) 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.

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. - $\text { If } A \cup C = B \cup C \text {, then } A = B \text {. }$

Find $\bigcap _ { i = 1 } ^ { + \infty } \left[ 1 - \frac { 1 } { i } , 1 \right]$

Suppose $f : \mathbf { R } \rightarrow \mathbf { Z }$ where $f ( x ) = \lceil 2 x - 1 \rceil$ (a) Draw the graph of $f$ (b) Is $f$ 1-1 ? (Explain) (c) Is $f$ onto Z ? (Explain)

Find the sum 2-4+8-16+32-...-2²⁸ .

Show that $\lceil x \rceil = - \lfloor - x \rfloor$

Verify that $a _ { n } = 7 \cdot 3 ^ { n } - \pi$ is a solution to the recurrence relation $a _ { n } = 4 a _ { n - 1 } - 3 a _ { n - 2 }$

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. - $a _ { n } = \text { the number of subsets of a set of size } n$

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

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - $\left\{ x \mid x \in \mathbf { N } \text { and } 9 x ^ { 2 } - 1 = 0 \right\}$

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

Find the sum 2+4+8+16+32+...+2²⁸ .

In questions determine whether the statement is true or false. - $\text { If } \mathbf { A B } = \mathbf { A C } \text {, then } \mathbf { B } = \mathbf { C } \text {. }$

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

Prove or disprove: For all positive real numbers $x$ and $y$ , $\lfloor x \cdot y \rfloor \leq \lfloor x \rfloor \cdot \lfloor y \rfloor$

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

Suppose B = $\left( \begin{array} { l l } 3 & 5 \\2 & 4\end{array} \right)$ 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.