Warm-up-exercises Let
x
,
y
,
z
,
w
{\displaystyle {}x,y,z,w}
z
{\displaystyle {}z}
w
{\displaystyle {}w}
x
1
=
x
,
{\displaystyle {}{\frac {x}{1}}=x\,,}
1
z
=
z
−
1
,
{\displaystyle {}{\frac {1}{z}}=z^{-1}\,,}
1
−
1
=
−
1
,
{\displaystyle {}{\frac {1}{-1}}=-1\,,}
0
z
=
0
,
{\displaystyle {}{\frac {0}{z}}=0\,,}
z
z
=
1
,
{\displaystyle {}{\frac {z}{z}}=1\,,}
x
z
=
x
w
z
w
{\displaystyle {}{\frac {x}{z}}={\frac {xw}{zw}}\,}
x
z
⋅
y
w
=
x
y
z
w
,
{\displaystyle {}{\frac {x}{z}}\cdot {\frac {y}{w}}={\frac {xy}{zw}}\,,}
x
z
+
y
w
=
x
w
+
y
z
z
w
.
{\displaystyle {}{\frac {x}{z}}+{\frac {y}{w}}={\frac {xw+yz}{zw}}\,.}
Does there exist an analogue of formula (8), which arises when one replaces addition by multiplication (and subtraction by division), that is
(
x
−
z
)
⋅
(
y
−
w
)
=
(
x
+
w
)
⋅
(
y
+
z
)
−
(
z
+
w
)
?
{\displaystyle {}(x-z)\cdot (y-w)=(x+w)\cdot (y+z)-(z+w)\,?}
Show that the popular formula
x
z
+
y
w
=
x
+
y
z
+
w
{\displaystyle {}{\frac {x}{z}}+{\frac {y}{w}}={\frac {x+y}{z+w}}\,}
does not hold.
Determine which of the two rational numbers
p
{\displaystyle {}p}
q
{\displaystyle {}q}
p
=
573
−
1234
and
q
=
−
2007
4322
.
{\displaystyle p={\frac {573}{-1234}}{\text{ and }}q={\frac {-2007}{4322}}.}
a) Give an example of rational numbers
a
,
b
,
c
∈
]
0
,
1
[
{\displaystyle {}a,b,c\in {]0,1[}}
a
2
+
b
2
=
c
2
.
{\displaystyle {}a^{2}+b^{2}=c^{2}\,.}
b) Give an example of rational numbers
a
,
b
,
c
∈
]
0
,
1
[
{\displaystyle {}a,b,c\in {]0,1[}}
a
2
+
b
2
≠
c
2
.
{\displaystyle {}a^{2}+b^{2}\neq c^{2}\,.}
c) Give an example of irrational numbers
a
,
b
∈
]
0
,
1
[
{\displaystyle {}a,b\in {]0,1[}}
c
∈
]
0
,
1
[
{\displaystyle {}c\in {]0,1[}}
a
2
+
b
2
=
c
2
.
{\displaystyle {}a^{2}+b^{2}=c^{2}\,.}
Prove the following properties of real numbers.
1
≥
0
{\displaystyle {}1\geq 0}
From
a
≥
b
{\displaystyle {}a\geq b}
c
≥
0
{\displaystyle {}c\geq 0}
a
c
≥
b
c
{\displaystyle {}ac\geq bc}
From
a
≥
b
{\displaystyle {}a\geq b}
c
≤
0
{\displaystyle {}c\leq 0}
a
c
≤
b
c
{\displaystyle {}ac\leq bc}
a
2
≥
0
{\displaystyle {}a^{2}\geq 0}
a
≥
b
≥
0
{\displaystyle {}a\geq b\geq 0}
a
n
≥
b
n
{\displaystyle {}a^{n}\geq b^{n}}
n
∈
N
{\displaystyle {}n\in \mathbb {N} }
From
a
≥
1
{\displaystyle {}a\geq 1}
a
n
≥
a
m
{\displaystyle {}a^{n}\geq a^{m}}
n
≥
m
{\displaystyle {}n\geq m}
From
a
>
0
{\displaystyle {}a>0}
1
a
>
0
{\displaystyle {}{\frac {1}{a}}>0}
From
a
>
b
>
0
{\displaystyle {}a>b>0}
1
a
>
1
b
{\displaystyle {}{\frac {1}{a}}>{\frac {1}{b}}}
Show that for
real numbers
x
≥
3
{\displaystyle {}x\geq 3}
x
2
+
(
x
+
1
)
2
≥
(
x
+
2
)
2
{\displaystyle {}x^{2}+(x+1)^{2}\geq (x+2)^{2}\,}
holds.
Let
x
<
y
{\displaystyle {}x<y}
x
+
y
2
{\displaystyle {}{\frac {x+y}{2}}}
x
<
x
+
y
2
<
y
{\displaystyle {}x<{\frac {x+y}{2}}<y\,}
hold.
Prove the following properties for the
absolute value function
R
⟶
R
,
x
⟼
|
x
|
,
{\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \vert {x}\vert ,}
(here let
x
,
y
{\displaystyle {}x,y}
|
x
|
≥
0
{\displaystyle {}\vert {x}\vert \geq 0}
|
x
|
=
0
{\displaystyle {}\vert {x}\vert =0}
x
=
0
{\displaystyle {}x=0}
|
x
|
=
|
y
|
{\displaystyle {}\vert {x}\vert =\vert {y}\vert }
x
=
y
{\displaystyle {}x=y}
x
=
−
y
{\displaystyle {}x=-y}
|
y
−
x
|
=
|
x
−
y
|
{\displaystyle {}\vert {y-x}\vert =\vert {x-y}\vert }
|
x
y
|
=
|
x
|
|
y
|
{\displaystyle {}\vert {xy}\vert =\vert {x}\vert \vert {y}\vert }
For
x
≠
0
{\displaystyle {}x\neq 0}
|
x
−
1
|
=
|
x
|
−
1
{\displaystyle {}\vert {x^{-1}}\vert =\vert {x}\vert ^{-1}}
We have
|
x
+
y
|
≤
|
x
|
+
|
y
|
{\displaystyle {}\vert {x+y}\vert \leq \vert {x}\vert +\vert {y}\vert }
|
x
+
y
|
≥
|
x
|
−
|
y
|
{\displaystyle {}\vert {x+y}\vert \geq \vert {x}\vert -\vert {y}\vert }
Sketch the following subsets of
R
2
{\displaystyle {}\mathbb {R} ^{2}}
{
(
x
,
y
)
∣
x
=
5
}
{\displaystyle {}{\left\{(x,y)\mid x=5\right\}}}
{
(
x
,
y
)
∣
x
≥
4
and
y
=
3
}
{\displaystyle {}{\left\{(x,y)\mid x\geq 4{\text{ and }}y=3\right\}}}
{
(
x
,
y
)
∣
y
2
≥
2
}
{\displaystyle {}{\left\{(x,y)\mid y^{2}\geq 2\right\}}}
{
(
x
,
y
)
∣
|
x
|
=
3
and
|
y
|
≤
2
}
{\displaystyle {}{\left\{(x,y)\mid \vert {x}\vert =3{\text{ and }}\vert {y}\vert \leq 2\right\}}}
{
(
x
,
y
)
∣
3
x
≥
y
and
5
x
≤
2
y
}
{\displaystyle {}{\left\{(x,y)\mid 3x\geq y{\text{ and }}5x\leq 2y\right\}}}
{
(
x
,
y
)
∣
x
y
=
0
}
{\displaystyle {}{\left\{(x,y)\mid xy=0\right\}}}
{
(
x
,
y
)
∣
x
y
=
1
}
{\displaystyle {}{\left\{(x,y)\mid xy=1\right\}}}
{
(
x
,
y
)
∣
x
y
≥
1
and
y
≥
x
3
}
{\displaystyle {}{\left\{(x,y)\mid xy\geq 1{\text{ and }}y\geq x^{3}\right\}}}
{
(
x
,
y
)
∣
0
=
0
}
{\displaystyle {}{\left\{(x,y)\mid 0=0\right\}}}
{
(
x
,
y
)
∣
0
=
1
}
{\displaystyle {}{\left\{(x,y)\mid 0=1\right\}}}
Hand-in-exercises Let
x
1
,
…
,
x
n
{\displaystyle {}x_{1},\ldots ,x_{n}}
|
∑
i
=
1
n
x
i
|
≤
∑
i
=
1
n
|
x
i
|
.
{\displaystyle {}\vert {\sum _{i=1}^{n}x_{i}}\vert \leq \sum _{i=1}^{n}\vert {x_{i}}\vert \,.}
Prove the general distributive property for a
field.
Sketch the following subsets of
R
2
{\displaystyle {}\mathbb {R} ^{2}}
{
(
x
,
y
)
∣
x
+
y
=
3
}
{\displaystyle {}{\left\{(x,y)\mid x+y=3\right\}}}
{
(
x
,
y
)
∣
x
+
y
≤
3
}
{\displaystyle {}{\left\{(x,y)\mid x+y\leq 3\right\}}}
{
(
x
,
y
)
∣
(
x
+
y
)
2
≥
4
}
{\displaystyle {}{\left\{(x,y)\mid (x+y)^{2}\geq 4\right\}}}
{
(
x
,
y
)
∣
|
x
+
2
|
≥
5
and
|
y
−
2
|
≤
3
}
{\displaystyle {}{\left\{(x,y)\mid \vert {x+2}\vert \geq 5{\text{ and }}\vert {y-2}\vert \leq 3\right\}}}
{
(
x
,
y
)
∣
|
x
|
=
0
and
|
y
4
−
2
y
3
+
7
y
−
5
|
≥
−
1
}
{\displaystyle {}{\left\{(x,y)\mid \vert {x}\vert =0{\text{ and }}\vert {y^{4}-2y^{3}+7y-5}\vert \geq -1\right\}}}
{
(
x
,
y
)
∣
−
1
≤
x
≤
3
and
0
≤
y
≤
x
3
}
{\displaystyle {}{\left\{(x,y)\mid -1\leq x\leq 3{\text{ and }}0\leq y\leq x^{3}\right\}}}
A page has been ripped off from a book. The sum of the numbers of the remaining pages is
65000
{\displaystyle {}65000}
![{\displaystyle {}a,b,c\in {]0,1[}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bf66fe1848171a59a1b5c61c59161f651b3ede)
![{\displaystyle {}a,b\in {]0,1[}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a8d5f76730252b7fbe1daec643ed47ad3102f98)
![{\displaystyle {}c\in {]0,1[}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01fea4cb646de38fb17e72f70c364d1e8b8d4ec2)
