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

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 animals living in the ocean, the set of ﬁsh.

(Short Answer)

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Question 121

Find $\bigcap _ { - 1 } ^ { + \infty } ( i , \infty ) .$

(Short Answer)

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Question 122

Find the sum 2 + 1/2 + 1/8 + 1/32 + 1/128 + · · · .

(Short Answer)

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Question 123

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

(True/False)

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Question 124

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

(Short Answer)

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Question 125

Give an example of a function f : N → Z that is both 1-1 and onto Z.

(Short Answer)

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Question 126

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

(True/False)

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Question 127

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

(True/False)

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Question 128

Suppose f : N → N has the rule f(n) = 4n + 1. Determine whether f is 1-1.

(Short Answer)

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Question 129

determine whether each of the following sets is countable or uncountable. For those that are countably inﬁnite exhibit a one-to-one correspondence between the set of positive integers and that set. -The set of positive rational numbers that can be written with denominators less than 3.

(Essay)

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Question 130

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

(True/False)

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Question 131

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $\text { There is a set } A \text { such that } | \mathcal { P } ( A ) | = 12 \text {. }$

(True/False)

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Question 132

determine whether the rule describes a function with the given domain and codomain. - $g : \mathbf { N } \rightarrow \mathbf { N } , \text { where } g ( n ) = \text { any integer } > n$

(Short Answer)

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Question 133

ﬁnd a recurrence relation with initial condition(s) satisﬁed by the sequence. Assume a0 is the ﬁrst term of the sequence. - $a _ { n } = 2 ^ { n }$

(Short Answer)

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Question 134

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

(Short Answer)

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Question 135

Prove that between every two rational numbers a/b and c/d (a) there is a rational number. (b) there are an inﬁnite number of rational numbers.

(Short Answer)

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Question 136

suppose g: A → B and f : B → C where A = {a, b, c, d}, B = {1, 2, 3}, C = {2, 3, 6, 8}, and g and f are deﬁned by g = {(a, 2), (b, 1), (c, 3), (d, 2)} and f = {(1, 8), (2, 3), (3, 2)}. - $\text { Find } f \circ g \text {. }$

(Short Answer)

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Question 137

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 people who were born in the U.S., the set of people who are U.S. citizens.

(Essay)

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Question 138

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

(True/False)

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Question 139

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

(Essay)

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Question 140

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

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 animals living in the ocean, the set of ﬁsh.

Find ⋂−1+∞(i,∞).\bigcap _ { - 1 } ^ { + \infty } ( i , \infty ) .⋂−1+∞​(i,∞).

Find the sum 2 + 1/2 + 1/8 + 1/32 + 1/128 + · · · .

suppose A = {x, y} and B = {x, {x}}. Mark the statement TRUE or FALSE. - ∅∈P(B)\emptyset \in \mathcal { P } ( B )∅∈P(B)

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

Give an example of a function f : N → Z that is both 1-1 and onto Z.

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

suppose 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 f : N → N has the rule f(n) = 4n + 1. Determine whether f is 1-1.

determine whether each of the following sets is countable or uncountable. For those that are countably inﬁnite exhibit a one-to-one correspondence between the set of positive integers and that set. -The set of positive rational numbers that can be written with denominators less than 3.

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

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - There is a set A such that ∣P(A)∣=12. \text { There is a set } A \text { such that } | \mathcal { P } ( A ) | = 12 \text {. } There is a set A such that ∣P(A)∣=12.

determine whether the rule describes a function with the given domain and codomain. - g:N→N, where g(n)= any integer >ng : \mathbf { N } \rightarrow \mathbf { N } , \text { where } g ( n ) = \text { any integer } > ng:N→N, where g(n)= any integer >n

ﬁnd a recurrence relation with initial condition(s) satisﬁed by the sequence. Assume a0 is the ﬁrst term of the sequence. - an=2na _ { n } = 2 ^ { n }an​=2n

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 \cap T Find S∩T

Prove that between every two rational numbers a/b and c/d (a) there is a rational number. (b) there are an inﬁnite number of rational numbers.

suppose g: A → B and f : B → C where A = {a, b, c, d}, B = {1, 2, 3}, C = {2, 3, 6, 8}, and g and f are deﬁned by g = {(a, 2), (b, 1), (c, 3), (d, 2)} and f = {(1, 8), (2, 3), (3, 2)}. - Find f∘g. \text { Find } f \circ g \text {. } Find f∘g.

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 people who were born in the U.S., the set of people who are U.S. citizens.

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

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.

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

Find $\bigcap _ { - 1 } ^ { + \infty } ( i , \infty ) .$

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

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

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

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

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

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $\text { There is a set } A \text { such that } | \mathcal { P } ( A ) | = 12 \text {. }$

determine whether the rule describes a function with the given domain and codomain. - $g : \mathbf { N } \rightarrow \mathbf { N } , \text { where } g ( n ) = \text { any integer } > n$

ﬁnd a recurrence relation with initial condition(s) satisﬁed by the sequence. Assume a0 is the ﬁrst term of the sequence. - $a _ { n } = 2 ^ { n }$

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

suppose g: A → B and f : B → C where A = {a, b, c, d}, B = {1, 2, 3}, C = {2, 3, 6, 8}, and g and f are deﬁned by g = {(a, 2), (b, 1), (c, 3), (d, 2)} and f = {(1, 8), (2, 3), (3, 2)}. - $\text { Find } f \circ g \text {. }$

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

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