master > master: notebook woche 2

notebooks/week-2.ipynb

@@ -6,7 +6,14 @@
    "source": [
     "# Woche 2 - 17.-23. Oktober 2022 #\n",
     "\n",
-    "Herzlich willkommen zur 2. Woche!"
+    "Herzlich willkommen zur 2. Woche!\n",
+    "\n",
+    "## Agenda ##\n",
+    "\n",
+    "- [x] Struktur\n",
+    "- [x] Aufwärme\n",
+    "- [x] Seminaraufgaben 2.x\n",
+    "- [ ] Besprechung von HA 1.x\n"
    ]
   },
   {
@@ -46,11 +53,11 @@
     "X = ['a', 'b', 'c', 'd', 'e'];\n",
     "Y = [2, 3, 5, 7, 11, 13, 17];\n",
     "Z = ['α', 'β', 'γ', 'δ'];\n",
-    "f = [('a', 5), ('b', 5), ('c', 17), ('d', 2)]\n",
+    "f = [('a', 5), ('b', 5), ('c', 17), ('d', 2), ('e', 5)]\n",
     "g = [(2, 'γ'), (3, 'α'), (5, 'β'), (7, 'δ'), (11, 'β'), (13, 'γ'), (17, 'β')];\n",
     "comp = Functions(\n",
-    "    Function(name=('$f$', 'X', 'Y'), domain=X, codomain=Y, fct=f),\n",
-    "    Function(name=('$g$', 'Y', 'Z'), domain=Y, codomain=Z, fct=g),\n",
+    "    Function(name=('f', 'X', 'Y'), domain=X, codomain=Y, fct=f),\n",
+    "    Function(name=('g', 'Y', 'Z'), domain=Y, codomain=Z, fct=g),\n",
     ");\n",
     "comp.draw(show_labels=True);"
    ]
@@ -63,14 +70,14 @@
    "source": [
     "A = np.random.randint(-100, 100, size=(4, 7));\n",
     "\n",
-    "X = randomset_alphabet(low=2, high=5);\n",
-    "Y = randomset_integers(low=2, high=5);\n",
-    "Z = randomset_greek(low=2, high=5);\n",
-    "f = random_function(X, Y);\n",
-    "g = random_function(Y, Z);\n",
+    "X = randomset_alphabet(N=3);\n",
+    "Y = randomset_integers(N=5);\n",
+    "Z = randomset_greek(N=7);\n",
+    "f = random_function(X, Y, injective=True);\n",
+    "g = random_function(Y, Z, injective=True);\n",
     "comp = Functions(\n",
-    "    Function(name=('$f$', 'X', 'Y'), domain=X, codomain=Y, fct=f),\n",
-    "    Function(name=('$g$', 'Y', 'Z'), domain=Y, codomain=Z, fct=g),\n",
+    "    Function(name=('f', 'X', 'Y'), domain=X, codomain=Y, fct=f),\n",
+    "    Function(name=('g', 'Y', 'Z'), domain=Y, codomain=Z, fct=g),\n",
     ");\n",
     "comp.draw(show_labels=False);"
    ]

src/maths/diagrams/sets.py

@@ -66,18 +66,24 @@ class Function(Generic[T1,T2]):
         return Functions(self).draw();
 
 class Functions:
+    name: str;
     fcts: list[Function];
 
     def __init__(self, *f: Function):
         self.fcts = list(f);
+        self.name = r' \circ '.join([ fct.name[0] for fct in self.fcts ][::-1]);
 
     def draw(self, show_labels: bool = True) -> Figure:
         N = len(self.fcts);
         obj = mplot.subplots(1, 1, constrained_layout=True);
         fig: Figure = obj[0];
         axs: Axes = obj[1];
-        axs.tick_params(axis='both', which='both', left=False, right=False, top=False, bottom=False, labelbottom=False, labelleft=False);
-        mplot.title('');
+        axs.tick_params(axis='both', which='both', labelbottom=False, labelleft=False);
+        mplot.title(f'Darstellung von ${self.name}$', fontdict={
+            'fontsize': 16,
+            'horizontalalignment': 'center',
+            'color': 'forestgreen',
+        });
         mplot.xlabel('');
         mplot.ylabel('');
         mplot.margins(x=MARGIN, y=MARGIN);
@@ -127,7 +133,7 @@ class Functions:
 
             for j, p in enumerate(p_codomain):
                 y = f.codomain[j];
-                marker = 'o' if (y in comp_range_next) else 'x';
+                marker = 'o' if (y in comp_range_next) else '.';
                 axs.scatter([p[0]], [p[1]], label='', color='black', marker=marker);
                 y_name = f.codomain[j];
                 if show_labels:
@@ -138,16 +144,16 @@ class Functions:
                 q = p_codomain[j];
                 x = f.domain[i];
                 if k == 0 or (x in comp_range):
-                    axs.plot([p[0], q[0]], [p[1], q[1]], label='', color='g', linewidth=2);
+                    axs.plot([p[0], q[0]], [p[1], q[1]], label='', color='black', linewidth=1);
                 else:
-                    axs.plot([p[0], q[0]], [p[1], q[1]], label='', color='g', linestyle='--', linewidth=1);
+                    axs.plot([p[0], q[0]], [p[1], q[1]], label='', color='red', linestyle='--', linewidth=1);
 
             anchor = anchors[k];
             fct_name, X_name, Y_name = f.name;
-            axs.annotate(text=f'{fct_name}', xy=anchor[0], ha='center', size=FONTSIZE_FCT);
+            axs.annotate(text=f'${fct_name}$', xy=anchor[0], ha='center', size=FONTSIZE_FCT);
             if k == 0:
-                axs.annotate(text=f'{X_name}', xy=anchor[1], ha='center', size=FONTSIZE_FCT);
-            axs.annotate(text=f'{Y_name}', xy=anchor[2], ha='center', size=FONTSIZE_FCT);
+                axs.annotate(text=f'${X_name}$', xy=anchor[1], ha='center', size=FONTSIZE_FCT);
+            axs.annotate(text=f'${Y_name}$', xy=anchor[2], ha='center', size=FONTSIZE_FCT);
             axs.add_patch(FancyArrowPatch(
                 anchor[3], anchor[4],
                 connectionstyle = 'arc3,rad=0.5',
@@ -176,8 +182,6 @@ def oval(
     P[-1, :] = P[0, :];
     return P;
 
-
-
 def random_points(
     nr_points: int,
     scale:     tuple[float, float] = (1., 1.),

src/maths/sets/random.py

@@ -34,16 +34,19 @@ T2 = TypeVar('T2');
 # METHODS
 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
-def randomset_integers(low: int, high: int) -> list[int]:
-    N = random.randint(low, high);
+def randomset_integers(N: int = -1, low: int = 1, high: int = 1) -> list[int]:
+    if N == -1:
+        N = random.randint(low, high);
     return list(range(1, N+1));
 
-def randomset_alphabet(low: int, high: int) -> list[int]:
-    N = random.randint(low, high);
+def randomset_alphabet(N: int = -1, low: int = 1, high: int = 1) -> list[int]:
+    if N == -1:
+        N = random.randint(low, high);
     return list([a for k, a in enumerate(ALPHA) if k < N]);
 
-def randomset_greek(low: int, high: int) -> list[int]:
-    N = random.randint(low, high);
+def randomset_greek(N: int = -1, low: int = 1, high: int = 1) -> list[int]:
+    if N == -1:
+        N = random.randint(low, high);
     return list([a for k, a in enumerate(GREEK) if k < N]);
 
 def random_function(
@@ -52,11 +55,39 @@ def random_function(
     injective: Optional[bool] = None,
     surjective: Optional[bool] = None,
 ) -> list[tuple[T1, T2]]:
-    # TODO: add feature to force injectivity/surjectivity, if possible.
-    # m = len(X);
-    # n = len(Y);
-    # if m > n:
-    #     injective = False;
-    # if m < n:
-    #     surjective = False;
-    return [ (x, random.choice(Y)) for x in X ];
+    m = len(X);
+    n = len(Y);
+    if m == 0:
+        return [];
+    if n == 0:
+        raise Exception(f'Impossible to create a function with {m} elements in the domain and {n} in the codomain.');
+    match (injective, surjective):
+        case (True, _):
+            assert m <= n, f'Impossible to create an injective function with {m} elements in the domain and {n} in the codomain.';
+            Y = random.sample(Y, m);
+            return [(x, y) for x, y in zip(X, Y)];
+        case (_, True):
+            assert m >= n, f'Impossible to create an surjective function with {m} elements in the domain and {n} in the codomain.';
+            indexes = random.sample(list(range(m)), n);
+            g = [ (indexes[j], Y[j]) for j in range(n) ] \
+                + [
+                    (i, random.choice(Y))
+                    for i in range(m)
+                    if not i in indexes
+                ];
+            g = sorted(g, key=lambda o: o[0]);
+            return [ (X[i], y) for (i, y) in g ];
+        case (False, _):
+            assert m > 1, f'Impossible to create a non-injective function with {m} elements in the domain.';
+            indexes = random.sample(list(range(m)), m);
+            g = random_function(indexes, Y);
+            [(i0, y0), (i1, y1)] = g[:2];
+            g[0] = (i0, y1);
+            g = sorted(g, key=lambda o: o[0]);
+            return [ (X[i], y) for (i, y) in g ];
+        case (_, False):
+            assert n > 1, f'Impossible to create a non-surjective function with {n} elements in the codomain.';
+            Y = random.sample(Y, n-1);
+            return random_function(X, Y);
+        case _:
+            return [ (x, random.choice(Y)) for x in X ];