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

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 41

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

(Short Answer)

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Question 42

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 43

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

(True/False)

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Question 44

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: - $f : \mathbf { Z } \times \mathbf { Z } \rightarrow \mathbf { Q } \text {, where } f ( m , n ) = m / n$

(Short Answer)

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Question 45

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - $A \times B \text {, where } A = \{ a , b , c \} \text { and } B = \emptyset$

(Short Answer)

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Question 46

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

(Short Answer)

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Question 47

Prove that there is no smallest positive rational number.

(Short Answer)

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Question 48

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

(True/False)

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Question 49

ﬁnd a recurrence relation with initial condition(s) satisﬁed by the sequence. Assume a0 is the ﬁrst term of the sequence. - $a _ { n } = 3 n - 1$

(Short Answer)

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Question 50

Determine whether f is a function from the set of all bit strings to the set of integers if f(S) is the position of a 1 bit in the bit string S .

(Essay)

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Question 51

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

(True/False)

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Question 52

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

(True/False)

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Question 53

describe each sequence recursively. Include initial conditions and assume that the sequences begin with $a _ { 1 }$ . - $1 ^ { 2 } , 2 ^ { 2 } , 3 ^ { 2 } , 4 ^ { 2 } , \ldots$

(Short Answer)

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Question 54

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

(Short Answer)

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Question 55

Suppose $f : \mathbf { R } \rightarrow \mathbf { Z }$ where $f ( x ) = \lceil 2 x - 1 \rceil$ . (a) If A= $\{ x \mid 1 \leq x \leq 4 \}$ , find $f ( A )$ . (b) If B={3,4,5,6,7} , find $f ( B )$ . (c) If C={-9,-8} , find $f ^ { - 1 } ( C )$ (d) If D={0.4,0.5,0.6} , find $f ^ { - 1 } ( D )$

(Short Answer)

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Question 56

determine whether the statement is true or false. -If AB = AC, then B = C.

(True/False)

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Question 57

describe each sequence recursively. Include initial conditions and assume that the sequences begin with $a _ { 1 }$ . -3, 2, 1, 0, −1, −2, . . .

(Short Answer)

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Question 58

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

(Short Answer)

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Question 59

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

(Short Answer)

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Question 60

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

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.

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

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 .

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

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - A×B, where A={a,b,c} and B=∅A \times B \text {, where } A = \{ a , b , c \} \text { and } B = \emptysetA×B, where A={a,b,c} and B=∅

suppose g: A → B and f : B → C where A = {1, 2, 3, 4}, B = {a, b, c}, C = {2, 8, 10}, and g and f are defined by g = f(1, b), (2, a), (3, b), (4, a)g and f = {(a, 8), (b, 10); (c, 2)}. - Find f∘f−1. \text { Find } f \circ f ^ { - 1 } \text {. } Find f∘f−1.

Prove that there is no smallest positive rational number.

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

ﬁnd a recurrence relation with initial condition(s) satisﬁed by the sequence. Assume a0 is the ﬁrst term of the sequence. - an=3n−1a _ { n } = 3 n - 1an​=3n−1

Determine whether f is a function from the set of all bit strings to the set of integers if f(S) is the position of a 1 bit in the bit string S .

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

suppose A = {1, 2, 3, 4, 5}. Mark the statement TRUE or FALSE. - {2,4}∈A×A\{ 2,4 \} \in A \times A{2,4}∈A×A

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1a _ { 1 } a1​ . - 12,22,32,42,…1 ^ { 2 } , 2 ^ { 2 } , 3 ^ { 2 } , 4 ^ { 2 } , \ldots12,22,32,42,…

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

determine whether the statement is true or false. -If AB = AC, then B = C.

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1a _ { 1 } 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=2}\left\{ x \mid x \in Z \text { and } x ^ { 2 } = 2 \right\}{x∣x∈Z and x2=2}

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

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

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

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

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - $A \times B \text {, where } A = \{ a , b , c \} \text { and } B = \emptyset$

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

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

ﬁnd a recurrence relation with initial condition(s) satisﬁed by the sequence. Assume a0 is the ﬁrst term of the sequence. - $a _ { n } = 3 n - 1$

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

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

describe each sequence recursively. Include initial conditions and assume that the sequences begin with $a _ { 1 }$ . - $1 ^ { 2 } , 2 ^ { 2 } , 3 ^ { 2 } , 4 ^ { 2 } , \ldots$

describe each sequence recursively. Include initial conditions and assume that the sequences begin with $a _ { 1 }$ . -3, 2, 1, 0, −1, −2, . . .

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