Lösungen

1

Sum[k, {k, 1, 8999}]

Sum[k, {k, 1, 9000}]

2  Sum[k, {k, 1, 8999}] + 9000

2

a

Remove[x]

{a, b, c, ab, ac, bc, abc} = {32, 36, 28, 17, 19, 21, 44}

s2 = 
      Solve[abc == a + b + c - ab - ac - bc + x, x] // Flatten

x = x /. s2

b

x1 = a - ab - ac + x

x1 = a - ab - ac + x

x2 = b - ab - bc + x

x2 = c - bc - ac + x

c

Aufgabe b: Unzulässiger negativer Wert

3

Remove[x, y]

MatrixForm[{{x, y, "¬x", "¬x v y"}, {0, 0, 1, 1}, {0, 
          1, 1, 1}, {1, 0, 0, 0}, {1, 1, 0, 1}}]

MatrixForm[{{x, y, "¬y", "x ^ ¬y"}, {0, 0, 1, 0}, {0, 
          1, 0, 0}, {1, 0, 1, 1}, {1, 1, 0, 0}}]

MatrixForm[{{x, 
          y, "¬x v y", "x ^ ¬y", "¬x v y ==> ¬x \
^ y"}, {0, 0, 1, 0, 0}, {0, 1, 1, 0, 0}, {1, 0, 0, 1, 1}, {1, 1, 1, 0, 
          0}}]

4

a

Remove[a, x, c]

x1 = {1, 4}; x2 = {2, 4}; x3 = {2, 1}; x4 = {1, 2}; x5 = {3, 3}; 
    x6 = {3, 1}; x7 = {4, 2}; c[x_] := Circle[x + {0, 0.2}, 0.2]

<<Graphics`Arrow`

a[x1_, x2_] := Arrow[x1, x2];
Show[
        Graphics[{a[x1, x2], c[x2], a[x3, x2], a[x1, x3], c[x1], a[x3, x1], 
            c[x5], a[x5, x4], a[x6, x5], a[x7, x6], a[x4, x7], 
            c[x7]}]];

b

Alle Kringel anfügen, alle Pfeil transitiv ergänzen und umkehren

c

Nein (wegen Kringel oder umgekehrten Pfeilen)

5

<< Algebra`InequalitySolve`

InequalitySolve[
      x + y ≥ -4  &&  -x + 
            y ≥ -2  &&  y ≤ 4, {x, 
        y}]

<< Graphics`InequalityGraphics`

InequalityPlot[
        x + y ≥ -4  &&  -x + 
              y ≥ -2  &&  y ≤ 4, {x, -9, 
          8}, {y, -5, 5}];

6

a

Remove[f, a, b, c]

f[x_] := a x^2 + b x  +  c;  
    g[x_] := Expand[(x + 3) (x + 5)] ;  f[x]==g[
        x]

Solve[{f[0] == g[0], f[1] == g[1], f[2] == g[2]}, {a, 
        b, c}]

b

sNew = Solve[g[x] == 0, {x}] // Flatten

x1 = x /. sNew[[1]];  
    x2 = x /. sNew[[2]];
{x1 + x2,  x1 x2}