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

determine whether the rule describes a function with the given domain and codomain. - $G : \mathbf { R } \rightarrow \mathbf { R } \text { where } G ( x ) = \left\{ \begin{array} { l l } x + 2 & \text { if } x \geq 0 \\x - 1 & \text { if } x \leq 4\end{array} \right.$

(Short Answer)

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Question 41

In questions determine whether the statement is true or false. - $\text { If } \mathbf { A } \text { is a } 6 \times 4 \text { matrix and } \mathbf { B } \text { is a } 4 \times 5 \text { matrix, then } \mathbf { A B } \text { has } 16 \text { entries. }$

(True/False)

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Question 42

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

(True/False)

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Question 43

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -1, 111, 11111, 1111111, . . . .

(Short Answer)

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Question 44

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . . -2, 0, 2, 0, 2, 0, 2, . . . .

(Short Answer)

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Question 45

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

(True/False)

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Question 46

In questions determine whether the statement is true or false. - $\text { If } \mathbf { A } = \left( \begin{array} { c c } 1 & 3 \\- 5 & 2\end{array} \right) , \text { then } \mathbf { A } ^ { 2 } = \left( \begin{array} { c c } 1 & 9 \\25 & 4\end{array} \right)$

(True/False)

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Question 47

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

(True/False)

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Question 48

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

(True/False)

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Question 49

Suppose A is a 6 X 8 matrix, B is an 8 X 5 matrix, and C is a 5 X 9 matrix. Find the number of rows, the number of columns, and the number of entries in A(BC).

(Short Answer)

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Question 50

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side).

(Essay)

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Question 51

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: Z × Z → Z where f(m, n) = m − n if m > n.

(Short Answer)

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Question 52

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

(True/False)

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Question 53

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

(True/False)

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Question 54

Let $f ( x ) = \left\lfloor x ^ { 3 } / 3 \right\rfloor$ Finf (S) S is: (a) {-2,-1,0,1,2,3} (b) {0,1,2,3,4,5} (c) }{1,5,7,11} . (d) }\{2,6,10,14}

(Short Answer)

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Question 55

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -The Fibonacci numbers.

(Short Answer)

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Question 56

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving an element table proof.

(Essay)

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Question 57

suppose the following are fuzzy sets: F=\{0.7,0.1 Bill, 0.8 Fran, 0.3 Olive, 0.5 Tom \}, R=\{0.4 Ann, 0.9 Bill, 0.9 Fran, 0.6 Olive, 0.7 Tom \}. - $\text { Find } F \cup R \text {. }$

(Short Answer)

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Question 58

ﬁnd the inverse of the function f or else explain why the function has no inverse. - $f : \mathbf { R } \rightarrow \mathbf { R } \text { where } f ( x ) = 3 x - 5$

(Short Answer)

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Question 59

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 60

determine whether the rule describes a function with the given domain and codomain. - $G : \mathbf { R } \rightarrow \mathbf { R } \text { where } G ( x ) = \left\{ \begin{array} { l l } x + 2 & \text { if } x \geq 0 \\x - 1 & \text { if } x \leq 4\end{array} \right.$

In questions determine whether the statement is true or false. - $\text { If } \mathbf { A } \text { is a } 6 \times 4 \text { matrix and } \mathbf { B } \text { is a } 4 \times 5 \text { matrix, then } \mathbf { A B } \text { has } 16 \text { entries. }$

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

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -1, 111, 11111, 1111111, . . . .

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . . -2, 0, 2, 0, 2, 0, 2, . . . .

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

In questions determine whether the statement is true or false. - $\text { If } \mathbf { A } = \left( \begin{array} { c c } 1 & 3 \\- 5 & 2\end{array} \right) , \text { then } \mathbf { A } ^ { 2 } = \left( \begin{array} { c c } 1 & 9 \\25 & 4\end{array} \right)$

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

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

Suppose A is a 6 X 8 matrix, B is an 8 X 5 matrix, and C is a 5 X 9 matrix. Find the number of rows, the number of columns, and the number of entries in A(BC).

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side).

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: Z × Z → Z where f(m, n) = m − n if m > n.

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

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

Let $f ( x ) = \left\lfloor x ^ { 3 } / 3 \right\rfloor$ Finf (S) S is: (a) {-2,-1,0,1,2,3} (b) {0,1,2,3,4,5} (c) }{1,5,7,11} . (d) }\{2,6,10,14}

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -The Fibonacci numbers.

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving an element table proof.

suppose the following are fuzzy sets: F=\{0.7,0.1 Bill, 0.8 Fran, 0.3 Olive, 0.5 Tom \}, R=\{0.4 Ann, 0.9 Bill, 0.9 Fran, 0.6 Olive, 0.7 Tom \}. - $\text { Find } F \cup R \text {. }$

ﬁnd the inverse of the function f or else explain why the function has no inverse. - $f : \mathbf { R } \rightarrow \mathbf { R } \text { where } f ( x ) = 3 x - 5$

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

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∩B=A∪B, then A=B\text { If } A \cap B = A \cup B , \text { then } A = B If A∩B=A∪B, then A=B

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -1, 111, 11111, 1111111, . . . .

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . . -2, 0, 2, 0, 2, 0, 2, . . . .

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⊆Bx \subseteq Bx⊆B

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

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

Suppose A is a 6 X 8 matrix, B is an 8 X 5 matrix, and C is a 5 X 9 matrix. Find the number of rows, the number of columns, and the number of entries in A(BC).

Prove that A∩B‾=Aˉ∪Bˉ\overline { A \cap B } = \bar { A } \cup \bar { B }A∩B=Aˉ∪Bˉ by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side).

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: Z × Z → Z where f(m, n) = m − n if m > n.

suppose A={a,b,c}A = \{ a , b , c \}A={a,b,c} Mark the statement TRUE or FALSE. - (c,c)∈A×A( c , c ) \in A \times A(c,c)∈A×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. - ∅∈P(B)\emptyset \in \mathcal { P } ( B )∅∈P(B)

Let f(x)=⌊x3/3⌋f ( x ) = \left\lfloor x ^ { 3 } / 3 \right\rfloorf(x)=⌊x3/3⌋ Finf (S) S is: (a) {-2,-1,0,1,2,3} (b) {0,1,2,3,4,5} (c) }{1,5,7,11} . (d) }\{2,6,10,14}

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -The Fibonacci numbers.

Prove that A∩B‾=Aˉ∪Bˉ\overline { A \cap B } = \bar { A } \cup \bar { B }A∩B=Aˉ∪Bˉ by giving an element table proof.

suppose the following are fuzzy sets: F=\{0.7,0.1 Bill, 0.8 Fran, 0.3 Olive, 0.5 Tom \}, R=\{0.4 Ann, 0.9 Bill, 0.9 Fran, 0.6 Olive, 0.7 Tom \}. - Find F∪R. \text { Find } F \cup R \text {. } Find F∪R.

ﬁnd the inverse of the function f or else explain why the function has no inverse. - f:R→R where f(x)=3x−5f : \mathbf { R } \rightarrow \mathbf { R } \text { where } f ( x ) = 3 x - 5f:R→R where f(x)=3x−5

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

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

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

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

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

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side).

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

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

Let $f ( x ) = \left\lfloor x ^ { 3 } / 3 \right\rfloor$ Finf (S) S is: (a) {-2,-1,0,1,2,3} (b) {0,1,2,3,4,5} (c) }{1,5,7,11} . (d) }\{2,6,10,14}

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving an element table proof.

suppose the following are fuzzy sets: F=\{0.7,0.1 Bill, 0.8 Fran, 0.3 Olive, 0.5 Tom \}, R=\{0.4 Ann, 0.9 Bill, 0.9 Fran, 0.6 Olive, 0.7 Tom \}. - $\text { Find } F \cup R \text {. }$

ﬁnd the inverse of the function f or else explain why the function has no inverse. - $f : \mathbf { R } \rightarrow \mathbf { R } \text { where } f ( x ) = 3 x - 5$

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