5 - Multiple quantifiers and negating sentences

Statements with multiple quantifiers

Order usually matters! If there is only a single type of quantifier order does not matter.

For example:

is not the same as:

Negation of statements

Upper bound/lower bound

Upper bound: $\ge$ every thing in the set
Lower bound: $\le$ every thing in the set

In class practice

$P(x)\wedge$ $\mathrm{\neg }Q(X)$

HW 5

Ben Finch

E 5.1
a) $x\le 2$ or $x\ge 3$
b) $x>3$ and $x\le 2$
c) $x>3$ and $x\ne 4$ and $x^2=16$
d) $3<x<4$ and $9\nless x^2\nless 16$
e) ($x=2$ or $x=3$) and $x^2-5x+6\ne 0$
f) For some $x\in \mathbb{R}$, it happens that $x^2+2x\le 0$
g) For every $x\in \mathbb{R}$, $x^2+2x\le 0$
h) There exists an $x\in \mathbb{R}$ such that for all $y\in \mathbb{R}$ it holds that $y\le x^2$
i) For every $x\in \mathbb{R}$, there exists some $y\in \mathbb{R}$ such that $y\le x^2$
j) There exists an $\epsilon >0$ such that for every $\delta >0$, there exists an $x\in \mathbb{R}$ such that $0<|x-2|<\delta$ and $|x^2-4|\ge \epsilon$

E 5.2
a) These two statements don't have the same truth value. The first is saying "for every $x$ in the real numbers, there exists some $y$ such that $x+y=0$" which is true. The second is saying, "there exists some $y$ in the real numbers, such that for each $x$ in the real numbers, $x+y=0$" which is false.
b) These two statements do have the same truth value, since both are true in all cases.
c) These two statements don't have the same truth value for similar reasons to part a. The first statement is true, but the second one is false.
d) These statements don't have the same truth value for similar reasons to parts a & c. The first statement is true, but the second one is false.

E 5.3
a) (3, 17). The upper bound is 17. It does not have a greatest element because for any number in the set, you can always find another number that is greater but still less than 17.

b) It is impossible for a set to have a greatest element and not have an upper bound. If a set has a greatest element, then that greatest element is by definition an upper bound of the set.

E 5.4
a) Yes, it is 1.
b) Yes, it is also 1.
c) Yes, it is 0.
d) No. You can always get a smaller element by infinitely increasing $n$.

E 5.5
a) Yes, for example 2.
b) No.
c) Yes, for example -1.
d) No.

E 5.6
a) $x\in S,\mathrm{\exists }y\in S,y>x$
b)$\mathrm{\exists }y\in S,y>x$
c) $\forall z\in \mathbb{R},\exists y\in S,y>z$