Problem exercises.ipynb

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    "import sympy as sym"
   ]
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   "source": [
    "sym.init_printing(use_latex = \"mathjax\")"
   ]
  },
  {
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   "metadata": {
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   "source": [
    "# Exercise problems"
   ]
  },
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     "slide_type": "skip"
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   "source": []
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   "source": [
    "Consider the following sets $$ U = \\left\\{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \\right\\} \\\\ A = \\left\\{ 1, 2, 3, 4, 5, \\right\\} \\\\ B = \\left\\{ 3, 4, 5, 6, 7 \\right\\} \\\\ C = \\left\\{ 7, 8 , 9 \\right\\} $$\n",
    "Calculate or show that $ A \\cap B \\\\ A \\cup B \\\\ B \\cap C \\\\ A \\cap C \\\\ A \\oplus B \\\\ A \\cup \\left( B \\cap C \\right) = \\left( A \\cup B \\right) \\cap \\left( A \\cup C \\right) $"
   ]
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     "data": {
      "text/latex": [
       "$$\\left\\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\\right\\}$$"
      ],
      "text/plain": [
       "{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U = sym.FiniteSet(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)\n",
    "U.is_UniversalSet\n",
    "U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left\\{1, 2, 3, 4, 5\\right\\}$$"
      ],
      "text/plain": [
       "{1, 2, 3, 4, 5}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
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   ],
   "source": [
    "A = sym.FiniteSet(1, 2, 3, 4, 5)\n",
    "A"
   ]
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left\\{3, 4, 5, 6, 7\\right\\}$$"
      ],
      "text/plain": [
       "{3, 4, 5, 6, 7}"
      ]
     },
     "execution_count": 5,
     "metadata": {},
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   ],
   "source": [
    "B = sym.FiniteSet(3, 4, 5, 6, 7)\n",
    "B"
   ]
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   "execution_count": 6,
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left\\{7, 8, 9\\right\\}$$"
      ],
      "text/plain": [
       "{7, 8, 9}"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = sym.FiniteSet(7, 8, 9)\n",
    "C"
   ]
  },
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   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left\\{3, 4, 5\\right\\}$$"
      ],
      "text/plain": [
       "{3, 4, 5}"
      ]
     },
     "execution_count": 7,
     "metadata": {},
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   ],
   "source": [
    "sym.Intersection(A, B)"
   ]
  },
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   "execution_count": 8,
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left\\{1, 2, 3, 4, 5, 6, 7\\right\\}$$"
      ],
      "text/plain": [
       "{1, 2, 3, 4, 5, 6, 7}"
      ]
     },
     "execution_count": 8,
     "metadata": {},
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   ],
   "source": [
    "sym.Union(A, B)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "slideshow": {
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left\\{7\\right\\}$$"
      ],
      "text/plain": [
       "{7}"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.Intersection(B, C)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\emptyset$$"
      ],
      "text/plain": [
       "∅"
      ]
     },
     "execution_count": 10,
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   ],
   "source": [
    "sym.Intersection(A, C)"
   ]
  },
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   "execution_count": 11,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left\\{1, 2, 6, 7\\right\\}$$"
      ],
      "text/plain": [
       "{1, 2, 6, 7}"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.Union(A, B) - sym.Intersection(A, B)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The symmetric difference, $ A \\oplus B $, above requires interpretation.  The lements $ 3, 5 $ are not included, so we have $ A \\oplus B = \\left\\{ 1, 2, 6, 7 \\right\\} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left\\{6, 7, 8, 9, 10\\right\\}$$"
      ],
      "text/plain": [
       "{6, 7, 8, 9, 10}"
      ]
     },
     "execution_count": 12,
     "metadata": {},
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   ],
   "source": [
    "A.complement(U)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "subslide"
    }
   },
   "source": [
    "As for the distributive law, $ A \\cup \\left( B \\cap C \\right) = \\left( A \\cup B \\right) \\cap \\left( A \\cup C \\right) $ we have:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
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   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left\\{1, 2, 3, 4, 5, 7\\right\\}$$"
      ],
      "text/plain": [
       "{1, 2, 3, 4, 5, 7}"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.Union(A, sym.Intersection(B, C))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left\\{1, 2, 3, 4, 5, 7\\right\\}$$"
      ],
      "text/plain": [
       "{1, 2, 3, 4, 5, 7}"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.Intersection(sym.Union(A, B), sym.Union(A, C))"
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
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    "collapsed": true
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