집합과 명제 전체 문제

Complete the following.

Write the contrapositive of the following statement: For all positive real numbers $a$ and $b$, if $\sqrt {ab} \ne \dfrac{{a + b}}{2}$, then $a \ne b$.

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.

For a right triangle, suppose that the hypotenuse has length $c$ feet and the lengths of the sides are $a$ feet and $b$ feet.

What is a formula for the area of this right triangle? What is an isosceles triangle?

For a right triangle, suppose that the hypotenuse has length $c$ feet and the lengths of the sides are $a$ feet and $b$ feet.

State the Pythagorean Theorem for right triangles.

For a right triangle, suppose that the hypotenuse has length $c$ feet and the lengths of the sides are $a$ feet and $b$ feet.

Prove that the right triangle described above is an isosceles triangle if and only if the area of the right triangle is $\dfrac{1}{4} c^2$.

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.

Prove that for each integer $a$, if $a^2 - 1$ is even, then 4 divides $a^2 - 1$.

Prove that for all integers $a$ and $m$, if $a$ and $m$ are the lengths of the sides of a right triangle and $m + 1$ is the length of the hypotenuse, then $a$ is an odd integer.

Prove the following proposition:

If

p, q \in \mathbb{Q}

with

p \lt q

, then there exists an

x \in \mathbb{Q}

with

p \lt x \lt q

.

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 $6x + 15y = 2$.

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

There exist integers $x$ and $y$ such that $6x + 15y = 9$.

Let $y_1 , y_2 , y_3 , y_4$ be real numbers. The mean, $\overline y$, of these four numbers is defined to be the sum of the four numbers divided by 4. That is,

$$\overline y = \frac{{y_1 + y_2 + y_3 + y_4 }}{4}$$

. Prove that there exists a $y_i$ with $1 \leq i \leq 4$ such that $y_i \geq \overline y$.

Hint: One way is to let $y_{\text{ max } }$ be the largest of $y_1 , y_2 , y_3 , y_4$.

Prove the following proposition:

Let

a

and

b

be integers with

a \ne 0

. If

a

does not divide

b

, then the equation

ax^3 + bx + \left( b + a \right) = 0

does not have a solution that is a natural number.

Hint: It may be necessary to factor a sum of cubes. Recall that

$$u^3 + v^3 = \left( u + v \right) \left( u^2 - uv + v^2 \right)$$

.

This exercise is intended to provide another rationale as to why a proof by contradiction works. Suppose that we are trying to prove that a statement $P$ is true. Instead of proving this statement, assume that we prove that the conditional statement “If $\neg P$, then $C$” is true, where $C$ is some contradiction. Recall that a contradiction is a statement that is always false.

In symbols, write a statement that is a disjunction and that is logically equivalent to $\neg P \to C$.

This exercise is intended to provide another rationale as to why a proof by contradiction works. Suppose that we are trying to prove that a statement $P$ is true. Instead of proving this statement, assume that we prove that the conditional statement “If $\neg P$, then $C$” is true, where $C$ is some contradiction. Recall that a contradiction is a statement that is always false.

Now explain why $P$ must be true if we prove that the negation of $P$ implies a contradiction.

Are the following statements true or false? Justify each conclusion.

For all integers $a$ and $b$, if $a$ is even and $b$ is odd, then $4$ does not divide $\left( a^2 + b^2 \right)$.

Are the following statements true or false? Justify each conclusion.

For all integers $a$ and $b$, if $a$ is even and $b$ is odd, then $6$ does not divide $\left( a^2 + b^2 \right)$.