Establish, for each
n
∈
N
{\displaystyle {}n\in \mathbb {N} }
R
⟶
R
,
x
⟼
x
n
,
{\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{n},}
is
injective
and/or
surjective .
Show that there exists a
bijection
between
N
{\displaystyle {}\mathbb {N} }
Z
{\displaystyle {}\mathbb {Z} }
Give examples of
mappings
φ
,
ψ
:
N
⟶
N
,
{\displaystyle \varphi ,\psi \colon \mathbb {N} \longrightarrow \mathbb {N} ,}
such that
φ
{\displaystyle {}\varphi }
injective ,
but not
surjective ,
and
ψ
{\displaystyle {}\psi }
Let
L
,
M
,
N
{\displaystyle {}L,M,N}
P
{\displaystyle {}P}
F
:
L
⟶
M
,
x
⟼
F
(
x
)
,
{\displaystyle F\colon L\longrightarrow M,x\longmapsto F(x),}
G
:
M
⟶
N
,
y
⟼
G
(
y
)
,
{\displaystyle G\colon M\longrightarrow N,y\longmapsto G(y),}
and
H
:
N
⟶
P
,
z
⟼
H
(
z
)
,
{\displaystyle H\colon N\longrightarrow P,z\longmapsto H(z),}
be
functions .
Show that
H
∘
(
G
∘
F
)
=
(
H
∘
G
)
∘
F
.
{\displaystyle {}H\circ (G\circ F)=(H\circ G)\circ F\,.}
Let
L
,
M
,
N
{\displaystyle {}L,M,N}
f
:
L
⟶
M
and
g
:
M
⟶
N
{\displaystyle f:L\longrightarrow M{\text{ and }}g:M\longrightarrow N}
be
functions
with their
composition
g
∘
f
:
L
⟶
N
,
x
⟼
g
(
f
(
x
)
)
.
{\displaystyle g\circ f\colon L\longrightarrow N,x\longmapsto g(f(x)).}
Show that if
g
∘
f
{\displaystyle {}g\circ f}
injective ,
then also
f
{\displaystyle {}f}
Calculate in the
polynomial ring
C
[
X
]
{\displaystyle {}\mathbb {C} [X]}
(
(
4
+
i
)
X
2
−
3
X
+
9
i
)
⋅
(
(
−
3
+
7
i
)
X
2
+
(
2
+
2
i
)
X
−
1
+
6
i
)
.
{\displaystyle ((4+{\mathrm {i} })X^{2}-3X+9{\mathrm {i} })\cdot ((-3+7{\mathrm {i} })X^{2}+(2+2{\mathrm {i} })X-1+6{\mathrm {i} }).}
Let
K
{\displaystyle {}K}
K
[
X
]
{\displaystyle {}K[X]}
K
{\displaystyle {}K}
deg
(
P
+
Q
)
≤
max
{
deg
(
P
)
,
deg
(
Q
)
}
,
{\displaystyle {}\operatorname {deg} \,(P+Q)\leq \max\{\operatorname {deg} \,(P),\,\operatorname {deg} \,(Q)\}\,,}
deg
(
P
⋅
Q
)
=
deg
(
P
)
+
deg
(
Q
)
.
{\displaystyle {}\operatorname {deg} \,(P\cdot Q)=\operatorname {deg} \,(P)+\operatorname {deg} \,(Q)\,.}
Show that in a
polynomial ring
over a
field
K
{\displaystyle {}K}
P
,
Q
∈
K
[
X
]
{\displaystyle {}P,Q\in K[X]}
P
Q
≠
0
{\displaystyle {}PQ\neq 0}
Evaluate the
polynomial
2
X
3
−
5
X
2
−
4
X
+
7
{\displaystyle 2X^{3}-5X^{2}-4X+7}
replacing the variable
X
{\displaystyle {}X}
complex number
2
−
5
i
{\displaystyle {}2-5{\mathrm {i} }}
Let
K
{\displaystyle {}K}
K
[
X
]
{\displaystyle {}K[X]}
K
{\displaystyle {}K}
P
∈
K
[
X
]
{\displaystyle {}P\in K[X]}
P
≠
0
{\displaystyle {}P\neq 0}
P
=
(
X
−
λ
1
)
μ
1
⋯
(
X
−
λ
k
)
μ
k
⋅
Q
{\displaystyle {}P=(X-\lambda _{1})^{\mu _{1}}\cdots (X-\lambda _{k})^{\mu _{k}}\cdot Q\,}
where
μ
j
≥
1
{\displaystyle {}\mu _{j}\geq 1}
Q
{\displaystyle {}Q}
λ
1
,
…
,
λ
k
{\displaystyle {}\lambda _{1},\ldots ,\lambda _{k}}
μ
1
,
…
,
μ
k
{\displaystyle {}\mu _{1},\ldots ,\mu _{k}}
Let
F
∈
C
[
X
]
{\displaystyle {}F\in \mathbb {C} [X]}
polynomial .
Prove that
F
{\displaystyle {}F}
Determine the smallest real number for which the
Bernoulli inequality
with exponent
n
=
3
{\displaystyle {}n=3}
Consider the set
M
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
}
{\displaystyle {}M=\{1,2,3,4,5,6,7,8\}}
φ
:
M
⟶
M
,
x
⟼
φ
(
x
)
,
{\displaystyle \varphi \colon M\longrightarrow M,x\longmapsto \varphi (x),}
defined by the following table
x
{\displaystyle {}x}
1
{\displaystyle {}1}
2
{\displaystyle {}2}
3
{\displaystyle {}3}
4
{\displaystyle {}4}
5
{\displaystyle {}5}
6
{\displaystyle {}6}
7
{\displaystyle {}7}
8
{\displaystyle {}8}
φ
(
x
)
{\displaystyle {}\varphi (x)}
2
{\displaystyle {}2}
5
{\displaystyle {}5}
6
{\displaystyle {}6}
1
{\displaystyle {}1}
4
{\displaystyle {}4}
3
{\displaystyle {}3}
7
{\displaystyle {}7}
7
{\displaystyle {}7}
Compute
φ
1003
{\displaystyle {}\varphi ^{1003}}
1003
{\displaystyle {}1003}
φ
{\displaystyle {}\varphi }
Prove that a strictly increasing function
f
:
R
⟶
R
{\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} }
is injective.
Let
L
,
M
,
N
{\displaystyle {}L,M,N}
f
:
L
⟶
M
and
g
:
M
⟶
N
{\displaystyle f:L\longrightarrow M{\text{ and }}g:M\longrightarrow N}
be
functions
with their
composite
g
∘
f
:
L
⟶
N
,
x
⟼
g
(
f
(
x
)
)
.
{\displaystyle g\circ f\colon L\longrightarrow N,x\longmapsto g(f(x)).}
Show that if
g
∘
f
{\displaystyle {}g\circ f}
surjective ,
then also
g
{\displaystyle {}g}
Compute in the
polynomial ring
C
[
X
]
{\displaystyle {}\mathbb {C} [X]}
(
(
4
+
i
)
X
3
−
i
X
2
+
2
X
+
3
+
2
i
)
⋅
(
(
2
−
i
)
X
3
+
(
3
−
5
i
)
X
2
+
(
2
+
i
)
X
+
1
+
5
i
)
.
{\displaystyle ((4+{\mathrm {i} })X^{3}-{\mathrm {i} }X^{2}+2X+3+2{\mathrm {i} })\cdot ((2-{\mathrm {i} })X^{3}+(3-5{\mathrm {i} })X^{2}+(2+{\mathrm {i} })X+1+5{\mathrm {i} }).}
Perform, in the polynomial ring
C
[
X
]
{\displaystyle {}\mathbb {C} [X]}
P
T
{\displaystyle {}{\frac {P}{T}}}
P
=
(
5
+
X
2
+
i
X
+
3
−
i
)
X
4
+
X
2
+
i
X
+
3
−
i
X
2
+
(
3
−
2
X
2
+
i
X
+
3
−
i
)
X
−
1
{\displaystyle {}P=(5+X^{2}+{\mathrm {i} }X+3-{\mathrm {i} })X^{4}+X^{2}+{\mathrm {i} }X+3-{\mathrm {i} }X^{2}+(3-2X^{2}+{\mathrm {i} }X+3-{\mathrm {i} })X-1\,}
and
T
=
X
2
+
i
X
+
3
−
i
.
{\displaystyle {}T=X^{2}+{\mathrm {i} }X+3-{\mathrm {i} }\,.}
![{\displaystyle {}\mathbb {C} [X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc5a91ede960f75df1b6b90029496a19f3b38d14)
![{\displaystyle {}K[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7861df9df3d7463835d927c4d0d4c27f5e2f8416)
![{\displaystyle {}P,Q\in K[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f268286ca05e4e4b663e348b39b7a20a444f9e6e)
![{\displaystyle \psi \colon K[X]\longrightarrow K,P\longmapsto P(a),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/319779f46760396ed47e3abaf1c07de176917858)
![{\displaystyle {}\mathbb {Q} [X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05854fb7f5090d865105d7297fa178c8eb802fbf)
![{\displaystyle {}P\in K[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4692e3ffcbd66e900f8975113ff3f32f1cabc21a)
![{\displaystyle {}F\in \mathbb {C} [X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22b7e6b13ba51c7aad4d356f76a85a078189e056)
![{\displaystyle {}P\in \mathbb {R} [X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bf6bcf64e16c3ac6b30ecd6577e72648ab2325a)
