Additional exercises 0

Exercise

Identify the hypothesis and conclusion in the following statements. For the 4th and 5th statements, rewrite them in a way that clarifies which part is hypothesis and which is conclusion.

1. If $x$ is a positive real number that satisfies $x^2-2x+1<0$, then $x<1$.
2. If $x$ is a real number, then either $x\le 0$, $x\ge 0$.
3. If there exists a real number $x$ such that $x^4+1=0$, then any real number is either an integer or a multiple of $\pi$.
4. The inverse of a irrational number is also irrational.
5. For each $x\in \mathbb{R}$ there exists $y\in \mathbb{R}$ such that $x=y^2$.

Exercise

The negation of various statements is described in the following table.

Statement	Negation
$P$ or $Q$	Not $P$ and not $Q$
$P$ and $Q$	Not $P$ or not $Q$
If $P$, then $Q$	$P$ and not $Q$
For all $x$, $P(x)$	There exist $x$ such that not $P(x)$
There exists $x$ such that $P(x)$	For every $x$, not $P(x)$

Think of day-to-day examples which illustrate the logic described in this table.

Exercise

Find the negation of the following statements

1. Some prime numbers are odd.
2. Nobody is lazy.
3. Some horses are black.
4. For every $x\in \mathbb{R}$, $x^2\ge 0$.
5. There exists $x\in \mathbb{R}$ such that $x^2=7$.
6. For each $x\in \mathbb{R}$ there exists $y\in \mathbb{R}$ such that $x=y^2$.
7. If $x,y\ge 0$ then $\sqrt{x}=\sqrt{y}$ implies that $x=y$.

Exercise

What does each of the following statements mean, and which of them are true?

1. For every positive number $x$, and every positive number $y$, we have $y^2=x$.
2. There exists a positive number $x$ such that for every positive number $y$, we have $y^2=x$.
3. There exists a positive number $x$, and there exists a positive number $y$, such that $y^2=x$.
4. For every positive number $y$, there exists a positive number $x$ such that $y^2=x$.
5. There exists a positive number $y$ such that for every positive number $x$, we have $y^2=x$.

Exercise

Consider the following statement.

Let $f$ be differentiable on $(a,b)$ and suppose that $f$ takes a local minimum or maximum at the point $c\in (a,b)$. Then $f^{\prime }(c)=0$.

Discuss the possibility to improve the statement in the sense of making it stronger. Would this be stronger or weaker conclusion? Would this be a stronger or a weaker hypothesis?