집합과 명제 전체 문제

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For all integers $a$ and $b$ with $a \ne 0$, if $6 \mid \left(ab \right)$, then $6 \mid a$ or $6 \mid b$.

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For all integers $a$, $b$, and $c$ with $a \ne 0$, if $a$ divides $(b - 1)$ and $a$ divides $(c - 1)$, then $a$ divides $(bc - 1)$.

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For each integer $n$, if 7 divides $\left( n^2 - 4 \right)$, then 7 divides $(n - 2)$.

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For every integer $n$, $4n^2 + 7n + 6$ is an odd integer.

Hint: Make sure you first try some examples. How do you prove that an integer is an odd integer?

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For every odd integer $n$, $4n^2 + 7n + 6$ is an odd integer.

Hint: The following algebra may be useful.

$$4 \left( 2m + 1 \right)^2 + 7 \left( 2m + 1 \right) + 6 = 16m^2 + 30m + 17$$

.

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For all integers $a$, $b$, and $d$ with $d \ne 0$, if $d$ divides both $a-b$ and $a+b$, then $d$ divides $a$.

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For all integers $a$, $b$, and $c$ with $a \ne 0$, if $a \mid (bc)$, then $a \mid b$ or $a \mid c$.

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For each integer $a$, if there exists an integer $n$ such that $a$ divides $(8n + 7)$ and $a$ divides $(4n + 1)$, then $a$ divides 5.

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For each integer $a$, if there exists an integer $n$ such that $a$ divides $(9n + 5)$ and $a$ divides $(6n + 1)$, then $a$ divides 7.

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For each integer $n$, if $n$ is odd, then 8 divides $\left( n^4 + 4n^2 + 11 \right)$.

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For each integer $n$, if $n$ is odd, then 8 divides $\left( n^4 + n^2 + 2n \right)$.

Determine if each of the following statements is true or false. Provide a counterexample for statements that are false and provide a complete proof for those that are true.

For all real numbers $x$ and $y$, $\sqrt{xy} \leq \dfrac{x + y}{2}$.

Determine if each of the following statements is true or false. Provide a counterexample for statements that are false and provide a complete proof for those that are true.

For all real numbers $x$ and $y$, $xy \leq \left( \dfrac{x + y}{2} \right)^2$.

Determine if each of the following statements is true or false. Provide a counterexample for statements that are false and provide a complete proof for those that are true.

For all nonnegative real numbers $x$ and $y$, $\sqrt{xy} \leq \dfrac{x + y}{2}$.

Complete the following.

Is this statement true or false? Prove the statement if it is true or provide a counterexample if it is false.

Is the following proposition true or false?

For all integers

a

and

b

, if

ab

is even, then

a

is even or

b

is even.

Justify your conclusion by writing a proof if the proposition is true or by providing a counterexample if it is false.

A real number $x$ is defined to be a rational number

provided

$$\text{ there exist integers } m \text{ and } n \text{ with } n \ne 0 \text{ such that } x = \frac{m}{n}$$

. A real number that is not a rational number is called an irrational number.

It is known that if $x$ is a positive rational number, then there exist positive integers $m$ and $n$ with $n \ne 0$ such that $x = \dfrac{m}{n}$. Is the following proposition true or false? Explain.

For each positive real number

x

, if

x

is irrational, then

\sqrt x

is irrational.

Is the following proposition true or false? Justify your conclusion.

For each integer

n

,

n

is even if and only if 4 divides

n^2

.

Hint: Remember that there are two conditional statements associated with this biconditional statement. Be willing to consider the contrapositive of one of these conditional statements.

Are the following propositions true or false? Justify your conclusion.

There exist integers $x$ and $y$ such that $4x + 6y = 2$.

Are the following propositions true or false? Justify your conclusion.

There exist integers $x$ and $y$ such that $6x + 15y = 2$.