명제의 증명과 절대부등식

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

For all integers $a$ and $b$, if $a$ is odd and $b$ is odd, then $4$ divides $\left( a^2 + 3b^2 \right)$.

Consider the following statement:

For each positive real number

r

, if

r^2 = 18

, then

r

is irrational.

If you were setting up a proof by contradiction for this statement, what would you assume? Carefully write down all conditions that you would assume.

Prove that the cube root of 2 is an irrational number. That is, prove that if $r$ is a real number such that $r^3 = 2$, then $r$ is an irrational number.

Prove the following propositions:

For all real numbers $x$ and $y$, if $x$ is rational and $y$ is irrational, then $x + y$ is irrational.

Prove the following propositions:

For all nonzero real numbers $x$ and $y$, if $x$ is rational and $y$ is irrational, then $xy$ is irrational.

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

For each positive real number $x$, if $x$ is irrational, then $x^2$ is irrational.

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

For each positive real number $x$, if $x$ is irrational, then $\sqrt x$ is irrational.

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

For every pair of real numbers $x$ and $y$, if $x + y$ is irrational, then $x$ is irrational and $y$ is irrational.

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

For every pair of real numbers $x$ and $y$, if $x + y$ is irrational, then $x$ is irrational or $y$ is irrational.

Complete the following.

Give an example that shows that the sum of two irrational numbers can be a rational number.

Complete the following.

Now explain why the following proof that $\left( \sqrt{2} + \sqrt{5} \right)$ is an irrational number is not a valid proof: Since $\sqrt{2}$ and $\sqrt{5}$ are both irrational numbers, their sum is an irrational number. Therefore, $\left( \sqrt{2} + \sqrt{5} \right)$ is an irrational number.

Note: You may even assume that we have proven that $\sqrt{5}$ is an irrational number. (We have not proven this.)

Complete the following.

Is the real number $\sqrt 2 + \sqrt 5$ a rational number or an irrational number? Justify your conclusion.

Complete the following.

Prove that for each real number $x$, $\left(x + \sqrt{2} \right)$ is irrational or $\left(-x + \sqrt{2} \right)$ is irrational.

Complete the following.

Generalize the proposition in Part (a) for any irrational number (instead of just $\sqrt{2}$) and then prove the new proposition.

Is the following statement true or false?

For all positive real numbers

x

and

y

,

\sqrt {x + y} \leq \sqrt x + \sqrt y

.

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

For each real number

x

,

x \left( 1 - x \right) \leq \dfrac{1}{4}

.

Justify your conclusion for each of the following.

Is the base 2 logarithm of 32, $\log_2 (32)$, a rational number or an irrational number?

Justify your conclusion for each of the following.

Is the base 2 logarithm of 3, $\log _2 (3)$, a rational number or an irrational number?

Prove each of the following propositions:

For each real number $\theta$, if $0 \lt \theta \lt \dfrac{\textstyle \pi }{\textstyle 2}$, then $\left[ {\sin (\theta) + \cos (\theta) } \right] > 1$.

Hint: What happens if you expand $[ {\sin (\theta) + \cos (\theta) } ]^2$? Don't forget your trigonometric identities.

Prove each of the following propositions:

For all real numbers $a$ and $b$, if $a \ne 0$ and $b \ne 0$, then $\sqrt{a^2 + b^2} \ne a + b$.