2.Preliminaries.ipynb

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    "EE 502 P: Analytical Methods\n",
    "# 2: Mathematical Preliminaries\n",
    "## Prof. Eric Klavins, Electrical and Computer Engineering\n",
    "### 25 September, 2019\n",
    "Copyright &copy; 2019, University of Washington"
   ]
  },
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   "source": [
    "In this lecture we describe the fundamental objects of mathematics: Sets, relations, functions, and cardinality. Understanding these concepts are the key to mathematical literacy. For this lecture, we will follow the **chapter one** of the following textbook fairly closely:\n",
    "\n",
    "> [An Introduction to Real Analysis](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch1.pdf) by John K. Hunter,\n",
    "\n",
    "which is available for free online.\n",
    "\n",
    "Rather than repeating that chapter in this notebook, I will present my own take on the subject at the whiteboard using the following outline. \n",
    "\n",
    "- Sets\n",
    "  - Defs: sets, elements, the empty set\n",
    "  - Representation: finite, with ellipses, in words, set builder\n",
    "  - Examples\n",
    "    - Finite sets of numbers, strings, other sets, \n",
    "    - Naturals, integers, rationals, reals, complexes\n",
    "  - Subsets, Equality (in terms of subsets)\n",
    "  - The power set and an example\n",
    "  - Intersection, union, difference, complement (universe)\n",
    "  - Cartesian products, tuples\n",
    "  - Example: Dice, two dice, two dice summing to 7\n",
    "  - $\\mathbb{R}^n$ and vector notation\n",
    "  - Set proofs\n",
    "    - What is a proof?\n",
    "    - $ (A \\cap B) \\subseteq A$\n",
    "    - $A \\cup ( B \\cap C ) = (A \\cup B) \\cap (A \\cup C)$\n",
    "- Functions\n",
    "  - Definition (and synonyms). domain, codomain\n",
    "  - As a list of tuples, the identity function, characteristic function\n",
    "  - Example functions on $\\mathbb{Z}$\n",
    "  - Graph of a function as a subset of domain cross co-domain\n",
    "  - Range, surjection (onto), injection(1-1), bijection\n",
    "  - Note positive proofs vs negative counterexamples\n",
    "  - Examples: Which of the above are surjections, injections, and/or bijections?\n",
    "  - Composition and inverse, definition, domains, and codomains\n",
    "  - Examples from real numbers\n",
    "- Relations\n",
    "  - Definitions\n",
    "  - Examples\n",
    "    - List of tuples, \"has read\", has a common factor \n",
    "    - Function graphs are relations, but not conversely (in general)\n",
    "  - Order relations\n",
    "    - Less than, subset, divides\n",
    "  - Equivalence relations\n",
    "    - reduces to (for fractions), equal mod\n",
    "- Cardinality, and types of infinity\n",
    "  - Countable\n",
    "    - Definition\n",
    "    - Examples: Even numbers, rationals\n",
    "  - Uncountable\n",
    "    - Reals\n",
    "    - [0,1]\n",
    "    - Binary sequences\n",
    "    - Functions from {0,...,9} to {0,...,9}\n",
    "  - Uncomputable functions\n",
    "\n",
    "I will then discuss how to represent sets, relations, functions and cardinality in Python using the remainder of this notebook. "
   ]
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   "source": [
    "import math\n",
    "import sympy as sm\n",
    "sm.init_printing(use_latex='mathjax')\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.patches as pch\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Sets"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Python has a built in set object, which is documented here:\n",
    "\n",
    "> https://docs.python.org/2/library/stdtypes.html#set\n",
    "\n",
    "It is intended for representing finite sets of objects. Note that these objects are *not* the same as lists. There is no implicit ordering and each element can only show up once. \n",
    "\n",
    "All of the standard relations and operations on sets are supported, including $\\in$, $\\subseteq$, $=$, $\\cap$, $\\cup$, and $-$. In addition, you can use `len` to get the size of the set.\n",
    "\n",
    "To construct sets, you can simply use curly braces. You can also create sets from lists and lists from sets, using `set(L)` and `list(S)` where `L` is a list and `S` is a set.\n",
    "\n",
    "Sets can be constructed with comprehensions, just like lists. Use curly braces instead of square brackets. \n",
    "\n",
    "The empty set is constructed using `set()` (i.e. with no arguments). "
   ]
  },
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   "execution_count": 21,
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "({1, 2, 3}, set, 3)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Listing elements\n",
    "\n",
    "A = { 1, 2, 3, 3 }\n",
    "A, type(A), len(A)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 9,
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     "end_time": "2019-10-03T04:00:38.882313Z",
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    {
     "data": {
      "text/latex": [
       "$$\\left ( \\left\\{\\right\\}, \\quad 0\\right )$$"
      ],
      "text/plain": [
       "(set([]), 0)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The empty set\n",
    "\n",
    "E = {} \n",
    "E = set()\n",
    "E, len(E)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
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     "end_time": "2019-10-03T04:01:05.249737Z",
     "start_time": "2019-10-03T04:01:05.244353Z"
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   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left\\{1, 3, 5, 7, 9\\right\\}$$"
      ],
      "text/plain": [
       "set([1, 3, 5, 7, 9])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Using a set comprehension, which is like set builder notation\n",
    "\n",
    "B = { x for x in range(10) if x%2 != 0 }\n",
    "B"
   ]
  },
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    {
     "data": {
      "text/plain": [
       "(True, False)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Checking whether an element is in a set\n",
    "1 in A, 3 not in B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
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     "text": [
      "{1, 3}\n",
      "{1, 2, 3, 5, 7, 9}\n",
      "{2}\n",
      "False\n",
      "True\n",
      "True\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "({2, 5, 7, 9}, {2, 5, 7, 9}, True)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Set relations\n",
    "\n",
    "print(A.intersection(B))\n",
    "print(A.union(B))\n",
    "print(A.difference(B))\n",
    "print(A.isdisjoint(B))\n",
    "print({3,5}.issubset(B))\n",
    "print({3,5} <= B) # equivalent\n",
    "\n",
    "D1 = A.symmetric_difference(B)\n",
    "D2 = A.union(B).difference(A.intersection(B))\n",
    "D1, D2, D1 == D2"
   ]
  },
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   "cell_type": "code",
   "execution_count": 22,
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     "end_time": "2019-10-03T04:04:22.247182Z",
     "start_time": "2019-10-03T04:04:22.242460Z"
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   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left ( 1, \\quad \\left\\{2, 3\\right\\}\\right )$$"
      ],
      "text/plain": [
       "(1, set([2, 3]))"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Removing elements\n",
    "a = A.pop()\n",
    "a, A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left\\{2, 3, 12\\right\\}$$"
      ],
      "text/plain": [
       "set([2, 3, 12])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Adding elements\n",
    "\n",
    "A.add(12)\n",
    "A"
   ]
  },
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   "cell_type": "code",
   "execution_count": 24,
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     "end_time": "2019-10-03T04:04:34.751441Z",
     "start_time": "2019-10-03T04:04:34.745878Z"
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   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left ( \\left\\{3\\right\\}, \\quad \\left\\{2, 3, 12\\right\\}\\right )$$"
      ],
      "text/plain": [
       "(set([3]), set([2, 3, 12]))"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Copying\n",
    "\n",
    "C = A.copy()\n",
    "A.remove(12)\n",
    "A.discard(2)\n",
    "A, C"
   ]
  },
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   "cell_type": "code",
   "execution_count": 25,
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left\\{\\right\\}$$"
      ],
      "text/plain": [
       "set([])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C.clear()\n",
    "C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
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     "end_time": "2019-10-03T04:06:05.416014Z",
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   "source": [
    "# To do the cartesian product, you can define a function\n",
    "# Try: Define a function that takes a list of sets and returns their n-fold cartesian product\n",
    "\n",
    "def cartesian_product(X,Y):\n",
    "    return {(x,y) for x in X for y in Y}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:06:43.307393Z",
     "start_time": "2019-10-03T04:06:43.302290Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{(1, 3), (6, 6), (5, 6), (2, 1), (6, 2), (1, 6), (5, 1), (2, 5), (1, 2), (3, 3), (5, 5), (4, 4), (6, 3), (1, 5), (3, 6), (2, 2), (4, 1), (1, 1), (6, 4), (3, 2), (2, 6), (5, 4), (4, 5), (5, 2), (1, 4), (2, 3), (4, 2), (6, 5), (3, 5), (5, 3), (4, 6), (6, 1), (3, 1), (4, 3), (3, 4), (2, 4)} \n",
      "\n",
      "{(6, 1), (1, 6), (4, 3), (3, 4), (2, 5), (5, 2)}\n"
     ]
    }
   ],
   "source": [
    "# Dice example\n",
    "\n",
    "die = {1,2,3,4,5,6}\n",
    "twodice = cartesian_product(die,die)\n",
    "seven = { p for p in twodice if p[0] + p[1] == 7 }\n",
    "print(twodice, \"\\n\")\n",
    "print(seven)"
   ]
  },
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   "execution_count": 32,
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     "end_time": "2019-10-03T04:08:51.286969Z",
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   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "unhashable type: 'set'",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-32-ce8d6fb6fd57>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      4\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      5\u001b[0m \u001b[0mA\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m{\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m}\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 6\u001b[0;31m \u001b[0mS\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m{\u001b[0m \u001b[0mA\u001b[0m \u001b[0;34m}\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      7\u001b[0m \u001b[0mA\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0madd\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      8\u001b[0m \u001b[0mS\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mTypeError\u001b[0m: unhashable type: 'set'"
     ]
    }
   ],
   "source": [
    "# Sets of sets: This won't work becuase Python sets need to have immutable elements. \n",
    "# But with the definition below, if you had a variable pointing to {1,2} you could change it,\n",
    "# thereby changing the set.\n",
    "\n",
    "A = {1,2}\n",
    "S = { A }\n",
    "A.add(3)\n",
    "S"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:10:02.819135Z",
     "start_time": "2019-10-03T04:10:02.813819Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left\\{\\left\\{1, 2\\right\\}\\right\\}$$"
      ],
      "text/plain": [
       "set([frozenset([1, 2])])"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# To solve this problem, you use the built in type `frozenset`, which supports\n",
    "# all the functions that `set` does, except for those that modify the set.\n",
    "\n",
    "A = {1,2}\n",
    "S = { frozenset(A) }\n",
    "S"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Power Set\n",
    "\n",
    "The Power Set is also easy to define. This definition uses recursion. In particular,\n",
    "we use the fact that the power set of $S$ is equal to the set of sets resulting from removing\n",
    "a single element from $S$, unioned together with the their power sets. That is,\n",
    "\n",
    "$$\n",
    "\\mathcal{P}(S) = \\{ S \\} \\cup \\bigcup_{x \\in S} \\{ S - \\{x\\} \\} \\cup \\mathcal{P}(S - \\{x\\})\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:11:59.661140Z",
     "start_time": "2019-10-03T04:11:59.651476Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{1, 2, 3, 4}\n",
      "{1, 2}\n",
      "set()\n",
      "{1, 2, 4}\n",
      "{4}\n",
      "{1, 3, 4}\n",
      "{1, 3}\n",
      "{1, 2, 3}\n",
      "{2, 4}\n",
      "{3}\n",
      "{1, 4}\n",
      "{2, 3, 4}\n",
      "{3, 4}\n",
      "{2, 3}\n",
      "{1}\n",
      "{2}\n"
     ]
    }
   ],
   "source": [
    "def power_set(S):\n",
    "    P = set()\n",
    "    P.add(frozenset(S))\n",
    "    for x in S:\n",
    "        T = S.copy()\n",
    "        T.remove(x)\n",
    "        P = P.union(power_set(T))\n",
    "    return P\n",
    "\n",
    "S = {1,2,3,4}\n",
    "for s in power_set(S):\n",
    "    print(set(s))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Tuples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:13:43.367283Z",
     "start_time": "2019-10-03T04:13:43.360977Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "S = {(4, 5, 6), (1, 2, 3)}\n",
      "x = (1, 2, 3)\n",
      "x[0] = 1\n"
     ]
    }
   ],
   "source": [
    "# Tuples are a built in type in Python. You can treat them somewhat like lists.\n",
    "\n",
    "x = (1,2,3)\n",
    "y = (4,5,6)\n",
    "S = { x, y }\n",
    "print(\"S =\", S )\n",
    "print(\"x =\", x )\n",
    "print(\"x[0] =\", x[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
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     "end_time": "2019-10-03T04:13:43.979971Z",
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   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$2$$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Python supports multiple assignments in one step using tuples (or lists for that matter)\n",
    "\n",
    "a,b,c = x\n",
    "b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Relations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:14:32.586734Z",
     "start_time": "2019-10-03T04:14:32.581290Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(False, True)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Recall that a relation is just a set of tuples. \n",
    "\n",
    "R = {\n",
    "    (1,2),\n",
    "    (2,3),\n",
    "    (3,4)\n",
    "}\n",
    "\n",
    "# This is how you have to write 1 R 3 and 1 R 2\n",
    "(1,3) in R, (1,2) in R "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
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     "end_time": "2019-10-03T04:14:40.661229Z",
     "start_time": "2019-10-03T04:14:40.657018Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Of course, Python has all sorts of relations predefined\n",
    "\n",
    "1 < 2\n",
    "4 != 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:15:19.149905Z",
     "start_time": "2019-10-03T04:15:19.144861Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Since Python's relations are on (essentially) infinite sets, they are defined as functions. \n",
    "# You can also define relations in infinte sets. Such definitions take a pair of elements and\n",
    "# return true or false.\n",
    "\n",
    "def divides(x,y):\n",
    "    return x != y and x != 0 and y % x == 0\n",
    "\n",
    "divides(3,12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:16:06.990893Z",
     "start_time": "2019-10-03T04:16:06.971936Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left\\{\\left ( 1, \\quad 2\\right ), \\left ( 1, \\quad 3\\right ), \\left ( 1, \\quad 4\\right ), \\left ( 1, \\quad 5\\right ), \\left ( 1, \\quad 6\\right ), \\left ( 1, \\quad 7\\right ), \\left ( 1, \\quad 8\\right ), \\left ( 1, \\quad 9\\right ), \\left ( 1, \\quad 10\\right ), \\left ( 1, \\quad 11\\right ), \\left ( 1, \\quad 12\\right ), \\left ( 1, \\quad 13\\right ), \\left ( 1, \\quad 14\\right ), \\left ( 1, \\quad 15\\right ), \\left ( 1, \\quad 16\\right ), \\left ( 1, \\quad 17\\right ), \\left ( 1, \\quad 18\\right ), \\left ( 1, \\quad 19\\right ), \\left ( 2, \\quad 4\\right ), \\left ( 2, \\quad 6\\right ), \\left ( 2, \\quad 8\\right ), \\left ( 2, \\quad 10\\right ), \\left ( 2, \\quad 12\\right ), \\left ( 2, \\quad 14\\right ), \\left ( 2, \\quad 16\\right ), \\left ( 2, \\quad 18\\right ), \\left ( 3, \\quad 6\\right ), \\left ( 3, \\quad 9\\right ), \\left ( 3, \\quad 12\\right ), \\left ( 3, \\quad 15\\right ), \\left ( 3, \\quad 18\\right ), \\left ( 4, \\quad 8\\right ), \\left ( 4, \\quad 12\\right ), \\left ( 4, \\quad 16\\right ), \\left ( 5, \\quad 10\\right ), \\left ( 5, \\quad 15\\right ), \\left ( 6, \\quad 12\\right ), \\left ( 6, \\quad 18\\right ), \\left ( 7, \\quad 14\\right ), \\left ( 8, \\quad 16\\right ), \\left ( 9, \\quad 18\\right )\\right\\}$$"
      ],
      "text/plain": [
       "set([(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8), (1, 9), (1, 10), \n",
       "(1, 11), (1, 12), (1, 13), (1, 14), (1, 15), (1, 16), (1, 17), (1, 18), (1, 19\n",
       "), (2, 4), (2, 6), (2, 8), (2, 10), (2, 12), (2, 14), (2, 16), (2, 18), (3, 6)\n",
       ", (3, 9), (3, 12), (3, 15), (3, 18), (4, 8), (4, 12), (4, 16), (5, 10), (5, 15\n",
       "), (6, 12), (6, 18), (7, 14), (8, 16), (9, 18)])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ... and combined with set builder notation\n",
    "\n",
    "n = 20\n",
    "D = { (x,y) for x in range(1,n) for y in range(1,n) if divides(x,y)}\n",
    "D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:16:26.177909Z",
     "start_time": "2019-10-03T04:16:25.961111Z"
    }
   },
   "outputs": [
    {
     "data": {
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9L8eC3G3hrt+ALZlNrYffvo4gXJoAEZFC95BbKHTMhc1so1Swv9pZnPblRWhx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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# ... and plotted. The is a common way to draw finite relations, with arrows from a to b\n",
    "# for every (a,b) in R. \n",
    "\n",
    "points = [ \n",
    "    (\n",
    "        math.cos(2*math.pi*i / n), \n",
    "        math.sin(2*math.pi*i / n)\n",
    "    ) for i in range(1,n)\n",
    "]\n",
    "\n",
    "for i,p in enumerate(points):\n",
    "    if i != 0:\n",
    "        plt.text(1.1*p[0],1.1*p[1],i+1)\n",
    "    \n",
    "for x in range(1,n):\n",
    "    for y in range(1,n):\n",
    "        if x != 1 and (x,y) in D:\n",
    "            plt.arrow(\n",
    "                points[x-1][0], \n",
    "                points[x-1][1], \n",
    "                0.95*(points[y-1][0] - points[x-1][0]), \n",
    "                0.95*(points[y-1][1] - points[x-1][1]),\n",
    "                head_width=0.03,\n",
    "                length_includes_head=True)\n",
    "\n",
    "plt.xlim(-1,1)\n",
    "plt.ylim(-1,1)\n",
    "plt.axis(\"off\");\n",
    "plt.title(\"Divides relation on [2,19]\\n\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:19:22.252056Z",
     "start_time": "2019-10-03T04:19:22.245175Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left\\{2, 3, 5, 7, 11, 13, 17, 19\\right\\}$$"
      ],
      "text/plain": [
       "set([2, 3, 5, 7, 11, 13, 17, 19])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# A predecessor of y with respect to the relation R is the set { x | x R y }\n",
    "# The elements that only have 1 as predecessor in the relation are prime\n",
    "\n",
    "def preds(y):\n",
    "    return { x for x in range(n) if (x,y) in D }\n",
    "\n",
    "{ y for (x,y) in D if len(preds(y)) == 1 }"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:20:46.970456Z",
     "start_time": "2019-10-03T04:20:46.966040Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$2$$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# In Python you can represent functions with a finite numbner of elements as dictionaries\n",
    "\n",
    "f = { 1: 2, 2: 3 }\n",
    "f[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:21:08.188620Z",
     "start_time": "2019-10-03T04:21:08.183986Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left \\{ 1 : 3\\right \\}$$"
      ],
      "text/plain": [
       "{1: 3}"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Note that the dictionary format prevents you from mapping a single element to two elements in the range\n",
    "\n",
    "f = { 1: 2, 1: 3 }\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:21:37.808611Z",
     "start_time": "2019-10-03T04:21:37.801274Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left ( \\left\\{1, 2\\right\\}, \\quad \\left\\{2, 3\\right\\}\\right )$$"
      ],
      "text/plain": [
       "(set([1, 2]), set([2, 3]))"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# You can compute the domain and range of a function\n",
    "\n",
    "def domain(F):\n",
    "    return set(F.keys())\n",
    "\n",
    "def frange(F):\n",
    "    return set(F.values())\n",
    "\n",
    "f = {1: 2, 2: 3}\n",
    "\n",
    "domain(f), frange(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:22:28.148909Z",
     "start_time": "2019-10-03T04:22:28.141638Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left \\{ 1 : 20, \\quad 2 : 30\\right \\}$$"
      ],
      "text/plain": [
       "{1: 20, 2: 30}"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# You can write a simple function to compute the composition of two functions\n",
    "\n",
    "def compose(F, G):\n",
    "    H = {}\n",
    "    for x in domain(F):\n",
    "        H[x] = G[F[x]]\n",
    "    return H\n",
    "\n",
    "\n",
    "f = {1: 2, 2: 3}\n",
    "g = {2: 20, 3: 30}\n",
    "compose(f,g)  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Functions in Python\n",
    "\n",
    "Of course, functions in Python are defined using `def`. They can actually be treated as objects as well, and passed around as arguments, and so on.\n",
    "\n",
    "Another way to define functions is with a `lambda` expression, a concept available in most programming languages. This is used when you may not want to name the function, or if you need to write a function that returns another function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:24:04.010438Z",
     "start_time": "2019-10-03T04:24:04.007810Z"
    },
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def F(x):\n",
    "    return x + 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:24:25.691420Z",
     "start_time": "2019-10-03T04:24:25.686875Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$124$$"
      ],
      "text/plain": [
       "124"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Lambda example\n",
    "\n",
    "F = lambda x : x + 1\n",
    "\n",
    "F(123)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:25:05.872569Z",
     "start_time": "2019-10-03T04:25:05.868963Z"
    },
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Using this idea, we can convert a dictionary into a full fledged lambda function.\n",
    "\n",
    "def dict_to_function(F):\n",
    "    return lambda x : F[x] if x in domain(F) else None\n",
    "\n",
    "F = dict_to_function(f) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:25:15.273606Z",
     "start_time": "2019-10-03T04:25:15.235348Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(dict, function)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(f), type(F)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:25:43.752893Z",
     "start_time": "2019-10-03T04:25:43.748714Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left ( 2, \\quad 2\\right )$$"
      ],
      "text/plain": [
       "(2, 2)"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f[1], F(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-02T18:53:49.092862Z",
     "start_time": "2019-10-02T18:53:49.088512Z"
    }
   },
   "outputs": [],
   "source": [
    "FOG = dict_to_function(compose(f,g))\n",
    "FOG(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:27:47.838263Z",
     "start_time": "2019-10-03T04:27:47.834085Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$3$$"
      ],
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(lambda x, y : x + y)(1,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:28:05.626291Z",
     "start_time": "2019-10-03T04:28:05.622315Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "function"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(lambda x, y : x + y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Numbers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:29:07.498449Z",
     "start_time": "2019-10-03T04:29:07.493625Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The set Z of Integers is formally represented via sympy. But you can also just use\n",
    "# Python ints, but they have size limits.\n",
    "\n",
    "import sympy as sm\n",
    "\n",
    "sm.Integer(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:29:11.796857Z",
     "start_time": "2019-10-03T04:29:11.792550Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{1}{2}$$"
      ],
      "text/plain": [
       "1/2"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Rationals\n",
    "\n",
    "sm.Rational(1,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:29:14.084262Z",
     "start_time": "2019-10-03T04:29:14.079483Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\sqrt{2}$$"
      ],
      "text/plain": [
       "√2"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Algebraic and not rational\n",
    "\n",
    "sm.sqrt(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:29:20.721414Z",
     "start_time": "2019-10-03T04:29:20.717036Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left ( \\pi, \\quad e\\right )$$"
      ],
      "text/plain": [
       "(π, ℯ)"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Transcendental\n",
    "\n",
    "sm.pi, sm.E"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:29:53.490490Z",
     "start_time": "2019-10-03T04:29:53.486051Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Working with transcendental numbers\n",
    "# And my favorite equation\n",
    "\n",
    "sm.E**(sm.pi * sm.I) == -1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Interesting Sets"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-03T04:30:17.754418Z",
     "start_time": "2019-10-03T04:30:14.911935Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Python programs\n",
    "\n",
    "import ast\n",
    "\n",
    "def is_python(expr):\n",
    "    try:\n",
    "        ast.parse(expr)\n",
    "        return True\n",
    "    except:\n",
    "        return False\n",
    "    \n",
    "chars = '0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ!\"#$%&\\'()*+,-./:;<=>?@[\\\\]^_`{|}~ '\n",
    "strings = chars\n",
    "p = { expr for expr in s if is_python(expr) }\n",
    "\n",
    "strings = [s + c for s in strings for c in chars]\n",
    "p = p.union({ s for s in strings if is_python(s)})\n",
    "\n",
    "strings = [s + c for s in strings for c in chars]\n",
    "p = p.union({ s for s in strings if is_python(s)})\n",
    "\n",
    "\"f()\" in p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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   "codemirror_mode": {
    "name": "ipython",
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   "file_extension": ".py",
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