Lösungen / Solutions

Dt. Vers.

1

c = {8, 6};  a = {4, -2};  b = {-1, 3}; 
    d = {9, -7};

Solve[
      c == λ1 a +  μ1 b, {λ1, μ1}]

Solve[
      d == λ2 a +  μ2 b, {λ2, μ2}]

2

Remove[x, a, b]

a

Solve[4 (3  x + 9  b) - 2  a == 5  x + a - b, {x}]

b

Solve[
      2  b + 3 (4  x - c) == 4 (a - b + 3  x), {x}]

Solve[2  b + 3 (-c) == 4 (a - b), {x}]

3

Parallelität wie in der Ebene existiert nicht auf der Kugelsphäre. (Z.B. Grosskreise schneiden sich. Vektoren sind aber Klassen paralleler Pfeile.)

4

Programm (aus dem Logik-Script) / 
Programme (pris du script "logique")

\(\(Wtab[f_, m_] := 
        Module[{tabb, tabZ, res, newtabX, u, exttabX, output, 
            output1}, \ 
Attributes[
              f] = {Listable}; 
tabb[
              1] = {{0}, {1}}; 
tabb[u_] := 
            Flatten[Table[
                Append[\(tabb[u - 1]\)[\([k]\)], w], {k, 1, 
                  Length[tabb[u - 1]]}, {w, 0, 1}], 1]; 
          tabZ[u_] := \ tabb[u] /. {0 -> False, 1 -> True}; 
          res = Table[
              Apply[f, \(tabZ[m]\)[\([n]\)]], {n, 1, Length[tabZ[m]]}]; 
          newtabX = Transpose[tabZ[m]]; 
u = 
            Length[newtabX]; 
\ \ \ \ \ exttabX = 
            Transpose[
              Append[newtabX, res]]; 
\ \ \ \ \ Prepend[exttabX, 
                Append[Range[u], {expr}]] /. {True -> 1, False -> 0} // 
            MatrixForm];\)\)
sub[x_, y_] := Or[Not[x], y]; 
    Attributes[sub] = {Listable};
bij[x_, y_] := And[sub[x, y], sub[y, x]]; 
    Attributes[bij] = {Listable};
orExc[x_, y_] := Not[bij[x, y]]; 
    Attributes[bij] = {Listable};
asUsual[f_] := 
      StringReplace[
        ToString[
          TraditionalForm[f]], {"!" -> "¬"}]

Testen / Tester

f1a[x1_, x2_] := Not[Or[x1, x2]];
f1b[x1_, x2_] := Or[Not[x1], Not[x2]];
f1c[x1_, x2_] := orExc[x1, x2];
{f1a[x1, x2], f1b[x1, x2], f1c[x1, x2]} // asUsual

{Wtab[f1a,     2    ], Wtab[f1b,     2    ], 
      Wtab[f1c,     2    ]}

a

Remove[a, b];
f1[a_, b_] := Or[a,  Not[b]];   f1[a, b] // asUsual

Wtab[f1,   2]

b

Remove[a, b];
f2[a_, b_] := And[Or[a,  Not[b]], b];   f2[a, b] // 
      asUsual

Wtab[f2,   2]

c

Remove[a, b];
f3[a_, b_] := sub[And[Or[a,  Not[b]], b], a];   f3[a, b] // 
      asUsual

Wtab[f3,   2]

5

?Range

u = Union[Range[100], {"restN"}];
a = Range[6];  b = Range[3, 9]; c = Range[50, 100];

?Intersection

a

Intersection[a, Union[b, Complement[c]]]

b

Union[a, Intersection[Complement[b], c]]

Length[Union[a, Intersection[Complement[b], c]]]

6

a

Gruppe: Abgeschlossen (D(360)=D(0),....), assoz. da D(α) o D(β) = D(α+β), 
neutr = D(0), Inv.:    D(-α)=inv(D(α)). Sogar kommutativ: D(α+β)= D(β+α)

b

Zur Gruppe fehlt sicher das neutrale Element. Sx und Sy erzeugen aber eine Gruppe.

7

Remove[a, b, c, d, e, f, r]

A[1] = 0; A[2] = 2; A[3] = A[2] + 2*2;
A[4] = A[3] + 2*3; A[5] = A[4] + 2*4;
A[6] = A[5] + 2*5;
A[n_] := A[n - 1] + 2 (n - 1)

Table[A[k], {k, 1, 15}]

Table[A[k], {k, 1, 10}]/2

?Sum

B[n_] := 2 Sum[j - 1, {j, 1, n}]

Table[B[k], {k, 1, 15}]

Formel:

B[n]  // Simplify

Induktion:

Verankerung: O.k.

B[2]

Vererbungung von n auf n+1: O.k.

r[n_] := B[n - 1] + 2 (n - 1)  // Simplify

l[n_] := B[n] // Simplify

l[n] == r[n]

l[n + 1] == r[n + 1]