What the difference is between relations, operations, and functions

relation-operation-function.ipynb

{
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  {
   "cell_type": "markdown",
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   "source": [
    "# Relations\n",
    "\n",
    "Relations are a subset of a cartesian product of two sets (the domain and codomain)\n",
    "\n",
    "- They _might not_ have an arrow for every element in the domain (ie: `d`)\n",
    "- They can have _multiple_ arrows coming out of a single element in the domain to multiple elements in the codomain (ie: `b` `->` `2`, `3`)"
   ]
  },
  {
   "cell_type": "code",
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       "</g>\n",
       "<g id=\"clust2\" class=\"cluster\">\n",
       "<title>clusterC</title>\n",
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       "<text text-anchor=\"middle\" x=\"151\" y=\"-106.8\" font-family=\"Times,serif\" font-size=\"14.00\">codomain</text>\n",
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       "<!-- 3 -->\n",
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       "<title>3</title>\n",
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       "<text text-anchor=\"middle\" x=\"147\" y=\"-38.3\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
       "</g>\n",
       "<!-- b&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>b&#45;&gt;3</title>\n",
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       "</g>\n",
       "<!-- c -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>c</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"115\" cy=\"-156\" rx=\"27\" ry=\"18\"/>\n",
       "<text text-anchor=\"middle\" x=\"115\" y=\"-152.3\" font-family=\"Times,serif\" font-size=\"14.00\">c</text>\n",
       "</g>\n",
       "<!-- c&#45;&gt;3 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>c&#45;&gt;3</title>\n",
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       "</g>\n",
       "<!-- d -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>d</title>\n",
       "<ellipse fill=\"none\" stroke=\"blue\" cx=\"43\" cy=\"-156\" rx=\"27\" ry=\"18\"/>\n",
       "<text text-anchor=\"middle\" x=\"43\" y=\"-152.3\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
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   ],
   "source": [
    "from graphviz import Digraph\n",
    "\n",
    "g = Digraph('G')\n",
    "g.attr(compound='true')\n",
    "\n",
    "with g.subgraph(name='clusterD') as c:\n",
    "    c.attr(label='domain')\n",
    "    c.node('a')\n",
    "    c.node('b', color=\"blue\")\n",
    "    c.node('c')\n",
    "    c.node('d', color=\"blue\")\n",
    "\n",
    "with g.subgraph(name='clusterC') as c:\n",
    "    c.attr(label='codomain')\n",
    "    c.node('1')\n",
    "    with c.subgraph(name='clusterI') as c:\n",
    "        c.attr(label='image')\n",
    "        c.node('2', color=\"blue\")\n",
    "        c.node('3', color=\"blue\")\n",
    "\n",
    "g.edge('a', '2')\n",
    "g.edge('b', '2', color=\"blue\")\n",
    "g.edge('b', '3', color=\"blue\")\n",
    "g.edge('c', '3')\n",
    "    \n",
    "g"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Operations\n",
    "\n",
    "- More constrained than `relations`: a specific type of relation where _every_ element in the domain maps to some element in the codomain.\n",
    "- They can have multiple arrows coming out from a single element in the domain to multiple elements in the codomain (ie: `b` `->` `2`, `3`)\n",
    "- Common example of this behavior is the `√` operator, where every input has two outputs, positive and negative (ex: `√100` = `10` _and_ `-10`)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
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       "<text text-anchor=\"middle\" x=\"151\" y=\"-189.8\" font-family=\"Times,serif\" font-size=\"14.00\">domain</text>\n",
       "</g>\n",
       "<g id=\"clust2\" class=\"cluster\">\n",
       "<title>clusterC</title>\n",
       "<polygon fill=\"none\" stroke=\"black\" points=\"8,-8 8,-122 230,-122 230,-8 8,-8\"/>\n",
       "<text text-anchor=\"middle\" x=\"119\" y=\"-106.8\" font-family=\"Times,serif\" font-size=\"14.00\">codomain</text>\n",
       "</g>\n",
       "<g id=\"clust3\" class=\"cluster\">\n",
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       "</g>\n",
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       "<title>c</title>\n",
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       "<text text-anchor=\"middle\" x=\"115\" y=\"-152.3\" font-family=\"Times,serif\" font-size=\"14.00\">c</text>\n",
       "</g>\n",
       "<!-- c&#45;&gt;3 -->\n",
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       "<title>c&#45;&gt;3</title>\n",
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       "<!-- d -->\n",
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       "<title>d</title>\n",
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       "<text text-anchor=\"middle\" x=\"43\" y=\"-152.3\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
       "</g>\n",
       "<!-- d&#45;&gt;3 -->\n",
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       "<!-- 1 -->\n",
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       "<title>1</title>\n",
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     "execution_count": 2,
     "metadata": {},
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    }
   ],
   "source": [
    "from graphviz import Digraph\n",
    "\n",
    "g = Digraph('G')\n",
    "g.attr(compound='true')\n",
    "\n",
    "with g.subgraph(name='clusterD', comment='foo') as c:\n",
    "    c.attr(label='domain')\n",
    "    c.node('a')\n",
    "    c.node('b', color=\"blue\")\n",
    "    c.node('c')\n",
    "    c.node('d')\n",
    "\n",
    "with g.subgraph(name='clusterC') as c:\n",
    "    c.attr(label='codomain')\n",
    "    c.node('1')\n",
    "    with c.subgraph(name='clusterI') as c:\n",
    "        c.attr(label='image')\n",
    "        c.node('2', color=\"blue\")\n",
    "        c.node('3', color=\"blue\")\n",
    "\n",
    "g.edge('a', '2')\n",
    "g.edge('b', '2', color=\"blue\")\n",
    "g.edge('b', '3', color=\"blue\")\n",
    "g.edge('c', '3')\n",
    "g.edge('d', '3')\n",
    "\n",
    "g"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Functions\n",
    "\n",
    "- Type of relation where _every_ element in the domain maps to some element in the codomain.\n",
    "- More constrained than `operations`: unlike operations, an element in the domain can only have _one_ arrow going to an element in the codomain. However, multiple elements in the domain can point to the same element in the codomain (ie: `a`, `b`, `c` `->` `2`).\n",
    "- It may be one-to-one and/or onto.\n",
    "- The codomain and image might not be the same (unless it's a bijection). Some elements in the codomain may not be reached."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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       "<g id=\"node4\" class=\"node\">\n",
       "<title>d</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"79\" cy=\"-156\" rx=\"27\" ry=\"18\"/>\n",
       "<text text-anchor=\"middle\" x=\"79\" y=\"-152.3\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>3</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"115\" cy=\"-42\" rx=\"27\" ry=\"18\"/>\n",
       "<text text-anchor=\"middle\" x=\"115\" y=\"-38.3\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
       "</g>\n",
       "<!-- d&#45;&gt;3 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>d&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M84.47,-137.99C90.32,-119.79 99.67,-90.7 106.48,-69.49\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"109.87,-70.39 109.6,-59.8 103.21,-68.25 109.87,-70.39\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"43\" cy=\"-42\" rx=\"27\" ry=\"18\"/>\n",
       "<text text-anchor=\"middle\" x=\"43\" y=\"-38.3\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.dot.Digraph at 0x7f0c5c7be730>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from graphviz import Digraph\n",
    "\n",
    "g = Digraph('G')\n",
    "g.attr(compound='true')\n",
    "\n",
    "with g.subgraph(name='clusterD', comment='foo') as c:\n",
    "    c.attr(label='domain')\n",
    "    c.node('a', color=\"blue\")\n",
    "    c.node('b', color=\"blue\")\n",
    "    c.node('c', color=\"blue\")\n",
    "    c.node('d')\n",
    "\n",
    "with g.subgraph(name='clusterC') as c:\n",
    "    c.attr(label='codomain')\n",
    "    c.node('1')\n",
    "    with c.subgraph(name='clusterI') as c:\n",
    "        c.attr(label='image')\n",
    "        c.node('2', color=\"blue\")\n",
    "        c.node('3')\n",
    "\n",
    "g.edge('a', '2', color=\"blue\")\n",
    "g.edge('b', '2', color=\"blue\")\n",
    "g.edge('c', '2', color=\"blue\")\n",
    "g.edge('d', '3')\n",
    "\n",
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.2"
  }
 },
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