4 - Open Sentences

Open sentence

Example of an open sentence:

Let $x$ and $y$ be variables, both with the same domain R. We may define the open sentence

$P(x,y):x>y$.

You can combine open sentences with logical connectives

Quantifiers

The universal quantifier

$\forall$ is the universal quantifier; the expression $\forall x\in S,P(x)$ makes the assertion that the open sentence P(x) is universally true for any x in the domain S.

The existential quantifier

In class practice

1. $[\frac{3}{2},7]$
2. All real numbers
3. $x<7$
4. $(\frac{3}{2},7)$
5. $\mathrm{\varnothing }$

HW 4

Ben Finch

E 4.1

$P(x):x>1$,
$Q(x):x^2<16$
$R(x):x+1$ is even.

a) $S=\{2,3\}$
b) $S=\{-1,-3,1,3\}$
c) $S=\{x\in Z|x\le 2,x=2k$ for some $k$ in $\mathbb{Z}\}$

d) Logical equivalence:

Beginning statement:
$(P(x)\Rightarrow Q(x))\Rightarrow R(x)$

A => B is equivalent to ¬A ∨ B
$\neg (P(x)\Rightarrow Q(x))\vee R(x)$
$\neg (\neg P(x)\vee Q(x))\vee R(x)$

De Morgan's Laws
$(\neg \neg P(x)\wedge \neg Q(x))\vee R(x)$
$(P(x)\wedge \neg Q(x))\vee R(x)$

Therefore:
$S=$ $\{x\in Z|x>4$ or $x$ is odd$\}$

E 4.2
The statement "If x is an odd integer, then $\frac{3x+5}{2}$ is an odd integer" is logically equivalent to "x is an even integer or $\frac{3x+5}{2}$ is an odd integer"

1: false - 1 is not an even integer nor is $\frac{3(1)+5}{2}=4$ an odd integer.
2: true - 2 is an even integer.
3: true - $\frac{3(3)+5}{2}=7$ is an odd integer
4: true - 4 is an even integer
5: false - 5 is not an even integer nor is $\frac{3(5)+5}{2}=10$ an odd integer.
6: true - 6 is an even integer

The open statement $\forall x\in Z,P(x)$ is false since there are some values (for example 1 & 5) for which $P(x)$ is false.

E 4.3
The statement Q(x): If $x$ is an even integer, then $3x+5$ is an odd integer is logically equivalent to "$x$ is an odd integer or $3x+5$ is an odd integer"

1: true - 1 is an odd integer.
2: true - $3(2)+5=11$ is an odd integer
3: true - 3 is an odd integer
4: true - $3(4)+5=17$ is an odd integer
5: true - 5 is an odd integer
6: true - $6(4)+5=29$ is an odd integer

The statement $\forall x\in Z,Q(x)$ appears to be true based on the values tested.

E 4.4**
$\forall x\in U,(x\in A)\iff (x\in B)$

E 4.5

a) $\mathrm{\exists }x\in \mathbb{Z},4<x<6$
b) $\mathrm{\forall }x\in \mathbb{Z},x$ is odd $⟹$ $x^2$ is odd
c) $\mathrm{\forall }x\in \mathbb{Z},$ $x^2\in ODD$ $⟹$ $x\in ODD$
d $\mathrm{\forall }x\in \mathbb{R},x\notin \mathbb{Q}⟹x\ne 0$
e) $\forall x\in \mathbb{r},\forall y\in \mathbb{R},x\in \mathbb{Q}\wedge y\in \mathbb{Q}⟹(x+y)\in \mathbb{Q}$
f) $\mathrm{\forall }x\in \mathbb{R},x^2⟹(x>0\wedge x\in \mathbb{R})$
g) $\mathrm{\exists }x\in \mathbb{Z},x^2-5x+6=0$
h) $\mathrm{\forall }x\in \mathbb{R},x2-5x+6=0⟹x\in \mathbb{Z}$

E 4.6
a) There exists some value $x$ in the real numbers such that $x^2=2$
b) For all values $x$ that are integers, $x$ is even if and only if $x^2$ is even
c) For all values $x$ in the real numbers, if $x>1$ then $x^3>1$.
d) For all values $x$ in the real numbers, if $x^2-2x+1=0$ then $x=1$
e) There exists some value $x$ in the rational numbers such that $2x^3-x^2+2x-1=0$
f) For every real number $x$, there exists an integer $y$ and a number $z$ in the range 0 (inclusive) to 1 (not inclusive) such that $x$ is the sum of $y$ and $z$.