Question 1

$$\text { Show that the set of odd positive integers greater than } 3 \text { is countable. }$$

Accepted Answer

The function $f ( n ) = 2 n + 3$ is a one-to-one correspondence from the set of positive integers to the set of odd positive integers greater than 3 . Hence this set is countable.

Question 2

$$\text { Let } A , B \text {, and } C \text { be sets. Prove or disprove that } A - ( B \cap C ) = ( A - B ) \cup ( A - C ) \text {. }$$

Accepted Answer

We see that $A - ( B \cap C ) = A \cap \overline { B \cap C } = A \cap ( \bar { B } \cup \bar { C } ) = ( A \cap \bar { B } ) \cup ( A \cap \bar { C } ) = ( A - B ) \cup ( A - C )$. These equalities follow from the definition of the difference of two sets, De Morgan's law, the distributive law for intersection over union, and the definition of the difference of two sets, respectively.

Question 3

$$\text { Prove or disprove that } \boldsymbol { A } \boldsymbol { B } = \boldsymbol { B } \boldsymbol { A } \text { whenever } \boldsymbol { A } \text { and } \boldsymbol { B } \text { are } 2 \times 2 \text { matrices. }$$

Accepted Answer

This is false. Counterexamples are easy to find. For instance, let $\boldsymbol { A } = \left[ \begin{array} { l l } 0 & 0 \ 2 & 0 \end{array} \right]$ and $\boldsymbol { B } = \left[ \begin{array} { l l } 1 & 0 \ 1 & 0 \end{array} \right]$. Then $\boldsymbol { A } \boldsymbol { B } = \left[ \begin{array} { l l } 0 & 0 \ 2 & 0 \end{array} \right]$ while $\boldsymbol { B } \boldsymbol { A } = \left[ \begin{array} { l l } 0 & 0 \ 0 & 0 \end{array} \right]$

Question 4

Consider the function $f ( n ) = 2 \lfloor n / 2 \rfloor$ from $\mathbf { Z }$ to $\mathbf { Z }$. Is this function one-to-one? Is this function onto? Justify your answers.

Accepted Answer

Note that @#LAT-DLM& and @#LAT-DLM&. Hence @#LAT-DLM& is not

Question 5

$$\text { Find } \sum _ { j = 1 } ^ { 100 } 2 j + 5 \text { and } \sum _ { j = 5 } ^ { 100 } 3 ^ { j }$$

Accepted Answer

The answer of \[\text { Find } \sum _ {...

$\text { Show that the set of odd positive integers greater than } 3 \text { is countable. }$

$\text { Let } A , B \text {, and } C \text { be sets. Prove or disprove that } A - ( B \cap C ) = ( A - B ) \cup ( A - C ) \text {. }$

$\text { Prove or disprove that } \boldsymbol { A } \boldsymbol { B } = \boldsymbol { B } \boldsymbol { A } \text { whenever } \boldsymbol { A } \text { and } \boldsymbol { B } \text { are } 2 \times 2 \text { matrices. }$

Consider the function $f ( n ) = 2 \lfloor n / 2 \rfloor$ from $\mathbf { Z }$ to $\mathbf { Z }$ . Is this function one-to-one? Is this function onto? Justify your answers.

$\text { Find } \sum _ { j = 1 } ^ { 100 } 2 j + 5 \text { and } \sum _ { j = 5 } ^ { 100 } 3 ^ { j }$

The Foundations: Logic and Proofs

A: the Foundations: Logic and Proofs

A: 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

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

Show that the set of odd positive integers greater than 3 is countable. \text { Show that the set of odd positive integers greater than } 3 \text { is countable. } Show that the set of odd positive integers greater than 3 is countable.

Let A,B, and C be sets. Prove or disprove that A−(B∩C)=(A−B)∪(A−C). \text { Let } A , B \text {, and } C \text { be sets. Prove or disprove that } A - ( B \cap C ) = ( A - B ) \cup ( A - C ) \text {. } Let A,B, and C be sets. Prove or disprove that A−(B∩C)=(A−B)∪(A−C).

Consider the function f(n)=2⌊n/2⌋f ( n ) = 2 \lfloor n / 2 \rfloorf(n)=2⌊n/2⌋ from Z\mathbf { Z }Z to Z\mathbf { Z }Z . Is this function one-to-one? Is this function onto? Justify your answers.

Find ∑j=11002j+5 and ∑j=51003j\text { Find } \sum _ { j = 1 } ^ { 100 } 2 j + 5 \text { and } \sum _ { j = 5 } ^ { 100 } 3 ^ { j } Find ∑j=1100​2j+5 and ∑j=5100​3j