suppose the following are multisets: \[\begin{array} { c } S = \{ 6 \cdot a , 3 \cdot b , 2 \cdot c , 5 \cdot d \} \\ T = \{ 2 \cdot a , 4 \cdot b , 2 \cdot c \} \end{array}\] -\[\text { Find } S - T \text {. }\]

The answer of suppose the following are multisets: \[\begin{array} {...

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

describe each sequence recursively. Include initial conditions and assume that the sequences begin with $a _ { 1 }$ . -1, 101, 10101, 1010101, . . .

(Short Answer)

4.9/5

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Question 21

Prove or disprove: if A, B , and C are sets, then $A - ( B \cap C ) = ( A - B ) \cap ( A - C )$

(Short Answer)

4.8/5

(30)

Question 22

Find $\sum _ { j = 1 } ^ { 3 } \sum _ { i = 1 } ^ { 3 } i j .$

(Short Answer)

4.8/5

(32)

Question 23

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . . -1, 1/3, 1/5, 1/7, 1/9, . . .

(Short Answer)

4.7/5

(33)

Question 24

determine whether the rule describes a function with the given domain and codomain. - $F : \mathbf { R } \rightarrow \mathbf { R } , \text { where } F ( x ) = \frac { 1 } { x - 5 }$

(Short Answer)

4.7/5

(32)

Question 25

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

(Short Answer)

4.9/5

(30)

Question 26

Find the sum $2+4+8+16+32+\cdots+2^{28}$ .

(Short Answer)

4.8/5

(38)

Question 27

describe each sequence recursively. Include initial conditions and assume that the sequences begin with $a _ { 1 }$ . -1/2, 1/3, 1/4, 1/5, . . .

(Short Answer)

4.9/5

(29)

Question 28

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - $\left\{ x \mid x \in \mathbf { N } \text { and } 4 x ^ { 2 } - 8 = 0 \right\}$

(Short Answer)

4.9/5

(35)

Question 29

describe each sequence recursively. Include initial conditions and assume that the sequences begin with $a _ { 1 }$ . -an = n!

(Short Answer)

4.8/5

(36)

Question 30

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

(Short Answer)

4.7/5

(33)

Question 31

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

(True/False)

4.8/5

(35)

Question 32

suppose the following are multisets: S=\{6\cdota,3\cdotb,2\cdotc,5\cdotd\} T=\{2\cdota,4\cdotb,2\cdotc\} - $\text { Find } S + T \text {. }$

(Short Answer)

4.9/5

(36)

Question 33

suppose that g: A → B and f : B → C , where A = B = C = {1, 2, 3, 4}, g = {(1, 4), (2, 1), (3, 1), (4, 2)}, and f = {(1, 3), (2, 2), (3, 4), (4, 2)}. - $\text { Find } f \circ g \text {. }$

(Short Answer)

4.7/5

(29)

Question 34

suppose the following are multisets: S=\{6\cdota,3\cdotb,2\cdotc,5\cdotd\} T=\{2\cdota,4\cdotb,2\cdotc\} - $\text { Find } S - T \text {. }$

(Short Answer)

4.8/5

(26)

Question 35

For each of the pairs of sets to determine whether the ﬁrst is a subset of the second, the second is a subset of the ﬁrst, or neither is a subset of the other. -The set of students studying a programming language, the set of students studying Java.

(Essay)

4.8/5

(30)

Question 36

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

(True/False)

5.0/5

(36)

Question 37

describe each sequence recursively. Include initial conditions and assume that the sequences begin with $a _ { 1 }$ . - $a _ { n } = 5 ^ { n }$

(Short Answer)

4.9/5

(33)

Question 38

Let $f ( x ) = \lfloor x ^ { 3 } / 3 \rfloor$ . Find $f ( S )$ if $S$ 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)

4.9/5

(36)

Question 39

Find $A ^ { 2 }$ if A={1, a}

(Short Answer)

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(40)

Question 40

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

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1a _ { 1 } a1​ . -1, 101, 10101, 1010101, . . .

Prove or disprove: if A, B , and C are sets, then A−(B∩C)=(A−B)∩(A−C)A - ( B \cap C ) = ( A - B ) \cap ( A - C )A−(B∩C)=(A−B)∩(A−C)

Find ∑j=13∑i=13ij.\sum _ { j = 1 } ^ { 3 } \sum _ { i = 1 } ^ { 3 } i j .∑j=13​∑i=13​ij.

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . . -1, 1/3, 1/5, 1/7, 1/9, . . .

determine whether the rule describes a function with the given domain and codomain. - F:R→R, where F(x)=1x−5F : \mathbf { R } \rightarrow \mathbf { R } , \text { where } F ( x ) = \frac { 1 } { x - 5 }F:R→R, where F(x)=x−51​

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

Find the sum 2+4+8+16+32+⋯+2282+4+8+16+32+\cdots+2^{28}2+4+8+16+32+⋯+228 .

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1a _ { 1 } a1​ . -1/2, 1/3, 1/4, 1/5, . . .

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - {x∣x∈N and 4x2−8=0}\left\{ x \mid x \in \mathbf { N } \text { and } 4 x ^ { 2 } - 8 = 0 \right\}{x∣x∈N and 4x2−8=0}

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1a _ { 1 } a1​ . -an = n!

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

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

suppose the following are multisets: S=\{6\cdota,3\cdotb,2\cdotc,5\cdotd\} T=\{2\cdota,4\cdotb,2\cdotc\} - Find S+T. \text { Find } S + T \text {. } Find S+T.

suppose that g: A → B and f : B → C , where A = B = C = {1, 2, 3, 4}, g = {(1, 4), (2, 1), (3, 1), (4, 2)}, and f = {(1, 3), (2, 2), (3, 4), (4, 2)}. - Find f∘g. \text { Find } f \circ g \text {. } Find f∘g.

suppose the following are multisets: S=\{6\cdota,3\cdotb,2\cdotc,5\cdotd\} T=\{2\cdota,4\cdotb,2\cdotc\} - Find S−T. \text { Find } S - T \text {. } Find S−T.

For each of the pairs of sets to determine whether the ﬁrst is a subset of the second, the second is a subset of the ﬁrst, or neither is a subset of the other. -The set of students studying a programming language, the set of students studying Java.

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

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1a _ { 1 } a1​ . - an=5na _ { n } = 5 ^ { n }an​=5n

Let f(x)=⌊x3/3⌋f ( x ) = \lfloor x ^ { 3 } / 3 \rfloorf(x)=⌊x3/3⌋ . Find f(S)f ( S )f(S) if SSS 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} .

Find A2A ^ { 2 }A2 if A={1, a}

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

describe each sequence recursively. Include initial conditions and assume that the sequences begin with $a _ { 1 }$ . -1, 101, 10101, 1010101, . . .

Prove or disprove: if A, B , and C are sets, then $A - ( B \cap C ) = ( A - B ) \cap ( A - C )$

Find $\sum _ { j = 1 } ^ { 3 } \sum _ { i = 1 } ^ { 3 } i j .$

determine whether the rule describes a function with the given domain and codomain. - $F : \mathbf { R } \rightarrow \mathbf { R } , \text { where } F ( x ) = \frac { 1 } { x - 5 }$

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

Find the sum $2+4+8+16+32+\cdots+2^{28}$ .

describe each sequence recursively. Include initial conditions and assume that the sequences begin with $a _ { 1 }$ . -1/2, 1/3, 1/4, 1/5, . . .

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - $\left\{ x \mid x \in \mathbf { N } \text { and } 4 x ^ { 2 } - 8 = 0 \right\}$

describe each sequence recursively. Include initial conditions and assume that the sequences begin with $a _ { 1 }$ . -an = n!

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

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

suppose the following are multisets: S=\{6\cdota,3\cdotb,2\cdotc,5\cdotd\} T=\{2\cdota,4\cdotb,2\cdotc\} - $\text { Find } S + T \text {. }$

suppose that g: A → B and f : B → C , where A = B = C = {1, 2, 3, 4}, g = {(1, 4), (2, 1), (3, 1), (4, 2)}, and f = {(1, 3), (2, 2), (3, 4), (4, 2)}. - $\text { Find } f \circ g \text {. }$

suppose the following are multisets: S=\{6\cdota,3\cdotb,2\cdotc,5\cdotd\} T=\{2\cdota,4\cdotb,2\cdotc\} - $\text { Find } S - T \text {. }$

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

describe each sequence recursively. Include initial conditions and assume that the sequences begin with $a _ { 1 }$ . - $a _ { n } = 5 ^ { n }$

Let $f ( x ) = \lfloor x ^ { 3 } / 3 \rfloor$ . Find $f ( S )$ if $S$ 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} .

Find $A ^ { 2 }$ if A={1, a}