집합과 명제 전체 문제

The following exercises contain definitions or results from more advanced mathematics courses. Even though we may not understand all of the terms involved, it is still possible to recognize the structure of the given statements and write a meaningful negation of that statement.

In abstract algebra, a ring

consists of a nonempty set $R$ and two operations called addition and multiplication. A nonzero element $a$ in a ring $R$ is called a zero divisor

provided that there exists a nonzero element $b$ in $R$ such that $ab = 0$ or $ba = 0$. Carefully explain what it means to say that a nonzero element $a$ in a ring $R$ is not a zero divisor.

Give at least four different examples of integers $a$, $b$, and $c$ with $a \ne 0$ such that $a$ divides $b$ and $a$ divides $c$.

Construct a know-show table for the following proposition: For all integers $a$, $b$, and $c$ with $a \ne 0$, if $a$ divides $b$ and $a$ divides $c$, then $a$ divides $(b + c)$.

Use a counterexample to prove the following statement is false.

For all integers

a

and

b

, if 5 divides

a

or 5 divides

b

, then 5 divides

(5a + b)

.

Prove each of the following statements:

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

Prove each of the following statements:

For each integer $a$, if 4 divides $(a - 1)$, then 4 divides $\left( a^2 - 1 \right)$.

Hint: If 4 divides $(a - 1)$, then there exists an integer $k$ such that $a - 1 = 4k$ and so $a = 4k + 1$. Use algebra to rewrite $\left( a^2 - 1 \right) = (4k + 1)^2 - 1$.

For each of the following, use a counterexample to prove the statement is false.

For each odd natural number $n$, if $n > 3$, then 3 divides $\left( n^2 - 1 \right)$.

For each of the following, use a counterexample to prove the statement is false.

For each natural number $n$, $\left( 3 \cdot 2^n + 2 \cdot 3^n + 1 \right)$ is a prime number.

For each of the following, use a counterexample to prove the statement is false.

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

For each of the following, use a counterexample to prove the statement is false.

For each integer $a$, if 4 divides $\left( a^2 - 1 \right)$, then 4 divides $(a - 1)$.

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

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 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 $a \mid c$.

Complete the following.

If $x$ and $y$ are integers and $xy=1$, explain why $x=1$ or $x=-1$.

Prove the following proposition:

Let

a

be an integer. If there exists an integer

n

such that

a \mid \left(4n+3 \right)

and

a \mid \left( 2n+1 \right)

, then

a = 1

or

a = - 1

.

Hint: Use the fact that the only divisors of 1 are 1 and $-1$.

$\left( 4n + 3 \right) - 2 \left( 2n + 1 \right) = 1$.