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

You take a job that pays $25,000 annually. (a) How much do you earn n years from now if you receive a three percent raise each year? (b) How much do you earn n years from now if you receive a ﬁve percent raise each year? (c) How much do you earn n years from now if each year you receive a raise of $1000 plus two percent of your previous year's salary.

(Short Answer)

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Question 81

Suppose $g : A \rightarrow B$ and $f : B \rightarrow C$ , where $f \circ g$ is 1-1 and g is 1-1. Must $f$ be 1-1?

(Short Answer)

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Question 82

for each partial function, determine its domain, codomain, domain of deﬁnition, set of values for which it is undeﬁned or if it is a total function: - $f : \mathbf { Z } \rightarrow \mathbf { Z } \text {, where } f ( n ) = \lceil n / 2 \rceil$

(Short Answer)

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Question 83

for each partial function, determine its domain, codomain, domain of deﬁnition, set of values for which it is undeﬁned or if it is a total function: - $f : \mathbf { Z } \times \mathbf { Z } \rightarrow \mathbf { Z } , \text { where } f ( m , n ) = m - n \text { if } m > n$

(Short Answer)

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Question 84

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

(Short Answer)

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Question 85

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

(Short Answer)

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Question 86

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

(True/False)

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Question 87

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

(Short Answer)

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Question 88

determine whether the statement is true or false. -If A and B are 2 × 2 matrices, then A+B=B+A.

(True/False)

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Question 89

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

(True/False)

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Question 90

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 91

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

(True/False)

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Question 92

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

(True/False)

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Question 93

Suppose $f : \mathbf { R } \rightarrow \mathbf { R }$ where $f ( x ) = \lfloor x / 2\rfloor$ (a) If $S = \{ x \mid 1 \leq x \leq 6 \}$ , find $f ( S )$ (b) If T={3,4,5} , find $f ^ { - 1 } ( T )$

(Short Answer)

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Question 94

suppose A= $\left( \begin{array} { l l l } 1 & 0 & 1 \\0 & 1 & 1 \\1 & 1 & 0\end{array} \right)$ and B = $\left( \begin{array} { l l l } 0 & 1 & 0 \\0 & 1 & 1 \\1 & 0 & 0\end{array} \right)$ , Find (a) the join of A and B. (b) the meet of A and B. (c) the boolean product of A and B .

(Short Answer)

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Question 95

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

(Essay)

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Question 96

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

(Essay)

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Question 97

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

(True/False)

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Question 98

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

(Short Answer)

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Question 99

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . . -5, 9, 13, 17, 21, . . .

(Short Answer)

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Question 100

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

Suppose g:A→Bg : A \rightarrow Bg:A→B and f:B→Cf : B \rightarrow Cf:B→C , where f∘gf \circ gf∘g is 1-1 and g is 1-1. Must fff be 1-1?

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

Give an example of a function f:Z→Zf : \mathbb { Z } \rightarrow \mathbb { Z }f:Z→Z that is onto Z\mathbf { Z }Z but not 1-1.

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

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⌋

determine whether the statement is true or false. -If A and B are 2 × 2 matrices, then A+B=B+A.

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

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

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

Suppose f:R→Rf : \mathbf { R } \rightarrow \mathbf { R }f:R→R where f(x)=⌊x/2⌋f ( x ) = \lfloor x / 2\rfloorf(x)=⌊x/2⌋ (a) If S={x∣1≤x≤6}S = \{ x \mid 1 \leq x \leq 6 \}S={x∣1≤x≤6} , find f(S)f ( S )f(S) (b) If T={3,4,5} , find f−1(T)f ^ { - 1 } ( T )f−1(T)

Verify that an=3n+4a _ { n } = 3 ^ { n + 4 }an​=3n+4 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​

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​

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

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

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . . -5, 9, 13, 17, 21, . . .

The Foundations: Logic and Proofs

A: the Foundations: Logic and Proofs

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

Algorithms

A: Algorithms

Number Theory and Cryptography

A: Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: Advanced Counting Techniques

Relations

A: Relations

Graphs

A: Graphs

Trees

A: Trees

Boolean Algebra

A: Boolean Algebra

Modeling Computation

A: Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Filters

Suppose $g : A \rightarrow B$ and $f : B \rightarrow C$ , where $f \circ g$ is 1-1 and g is 1-1. Must $f$ be 1-1?

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

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

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

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

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

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

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

Suppose $f : \mathbf { R } \rightarrow \mathbf { R }$ where $f ( x ) = \lfloor x / 2\rfloor$ (a) If $S = \{ x \mid 1 \leq x \leq 6 \}$ , find $f ( S )$ (b) If T={3,4,5} , find $f ^ { - 1 } ( T )$

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

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

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

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