Implementation besides plotting is done. Proof for complement of prod…

…uct algebra is in the making

examples/product_spaces.ipynb

@@ -651,6 +651,58 @@
    "execution_count": 62,
    "outputs": []
   },
+  {
+   "metadata": {},
+   "cell_type": "markdown",
+   "source": [
+    "## Complement of the Product Algebra\n",
+    "\n",
+    "[This](https://www.math.ucdavis.edu/~hunter/m206/ch4_measure_notes.pdf) mentions that the complement of an element of the product measure is constructed by\n",
+    "$$\n",
+    "    (A \\times B)^c = (A^c \\times B) \\cup (A \\times B^c) \\cup (A^c \\times B^c).\n",
+    "$$\n",
+    "It is easy to see that this construction would produce exponential many elements with respect to the number of variables. This is unfortunate.\n",
+    "However, the correct complement can be formed with linear many terms, which is way more efficient. The following equations describe a proof by induction on how that can be done.\n",
+    "\n",
+    "Let\n",
+    "\\begin{align*}\n",
+    "    \\mathbb{A} &= A \\cup A^c \\, , \\\\\n",
+    "    \\mathbb{B} &= B \\cup B^c \\text{ and }\\\\\n",
+    "    \\mathbb{C} &= C \\cup C^c.\n",
+    "\\end{align*}\n",
+    "\n",
+    "### Induction Assumption\n",
+    "\n",
+    "\\begin{align*}\n",
+    "    (A \\times B)^c = (A^c \\times \\mathbb{B}) \\cup (A \\times B^C)\n",
+    "\\end{align*}\n",
+    "Proof:\n",
+    "\\begin{align*}\n",
+    "    (A \\times B)^c &= (A^c \\times B) \\cup (A \\times B^c) \\cup (A^c \\times B^c) \\\\\n",
+    "    &= (A^c \\times B) \\cup (A^c \\times B^c) \\cup (A \\times B^c) \\\\\n",
+    "    &= ( A^c \\times (B \\cup B^c) ) \\cup   (A \\times B^c) \\\\\n",
+    "    &= (A^c \\times \\mathbb{B}) \\cup (A \\times B^C) \\square\n",
+    "\\end{align*}\n",
+    "\n",
+    "### Induction Step\n",
+    "\n",
+    "\\begin{align*}\n",
+    "    (A \\times B \\times C)^c = (A^c \\times \\mathbb{B} \\times \\mathbb{C}) \\cup (A \\times B^C \\times \\mathbb{C} ) \\cup (A \\times B \\times C^c)\n",
+    "\\end{align*}\n",
+    "Proof:\n",
+    "\\begin{align*}\n",
+    "    (A \\times B \\times C)^c &= (A^c \\times B \\times C) \\cup (A \\times B^c \\times C) \\cup (A \\times B \\times C^c) \\cup \n",
+    "    (A^c \\times B^c \\times C) \\cup (A^c \\times B \\times C^c) \\cup (A \\times B^c \\times C^c) \\cup \n",
+    "    (A^c \\times B^c \\times C^c) \\\\\n",
+    "    &= (C \\times \\underbrace{(A^c \\times B) \\cup (A \\times B^c) \\cup (A^c \\times B^c))}_{\\text{Induction Assumption}} \\cup\n",
+    "    (C^c \\times  \\underbrace{(A^c \\times B) \\cup (A \\times B^c) \\cup (A^c \\times B^c))}_{\\text{Induction Assumption}} \\cup (A \\times B \\times C^c) \\\\\n",
+    "    &= (C \\times (A^c \\times \\mathbb{B}) \\cup (A \\times B^C)) \\cup \n",
+    "       (C^c \\times (A^c \\times \\mathbb{B}) \\cup (A \\times B^C)) \\cup (A \\times B \\times C^c)\\\\\n",
+    "    &= \n",
+    "\\end{align*}\n"
+   ],
+   "id": "511cdcad45f76bab"
+  },
   {
    "cell_type": "markdown",
    "source": [

requirements.txt

@@ -1,6 +1,4 @@
-portion~=2.4.2
 numpy~=1.26.1
 plotly~=5.20.0
 typing_extensions
-
 sortedcontainers~=2.4.0