Revisão teoria dos conjuntos

Revisão_lógica.ipynb

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   "source": [
    "# Revisão dos axiomas usando Python"
   ]
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    "### a) Axioma da existência do conjunto vazio\n",
    "\n",
    "# ∃B∀x(x ∉ B) ⇒ ∅\n",
    "\n",
    "* Para a definição de 'para todo (∀)' usarei um iterador[for] (uma maneira de percorrer todos os itens de um determinado conjunto)"
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     "text": [
      "set()\n",
      "{1}\n"
     ]
    }
   ],
   "source": [
    "B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\n",
    "\n",
    "print({x for x in {11, 12, 13} if x in B}) # Um conjunto vazio\n",
    "print({x for x in {1, 11, 12, 13} if x in B}) # Um conjunto com elementos"
   ]
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   "source": [
    "### b) Axioma da extencionalidade\n",
    "\n",
    "Se dois conjuntos tem os mesmos elementos, então, são o mesmo conjunto\n",
    "\n",
    "# ∀A∀B[∀x (x ∈ A x ∈ B) ⇒ A = B]"
   ]
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     "text": [
      "[True, True, True, True]\n",
      "True\n"
     ]
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   "source": [
    "A = {1, 2, 3, 4}\n",
    "B = {1, 2, 3, 4}\n",
    "\n",
    "print(list(map(lambda x: x in A and x in B, A))) # verifica elemento a elemento (∀)\n",
    "print(all(list(map(lambda x: x in A and x in B, A)))) # verifica se todas as condições são verdadeiras\n",
    "\n",
    "assert A == B # Logo, se todos os elementos são iguais, representam o mesmo conjunto"
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     "text": [
      "[False, True, True, True]\n"
     ]
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     "ename": "AssertionError",
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      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
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      "\u001b[0;31mAssertionError\u001b[0m: "
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   "source": [
    "# Um contra exemplo:\n",
    "C = {2, 3, 4, 5}\n",
    "print(list(map(lambda x: x in C and x in B, B))) # verifica elemento a elemento (∀)\n",
    "all(list(map(lambda x: x in C and x in B, A))) # verifica se todas as condições são verdadeiras\n",
    "\n",
    "assert C == B # Logo, se todos os elementos são iguais, representam o mesmo conjunto"
   ]
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   "source": [
    "### c) Esquema do Ax. da compreensão\n",
    "\n",
    "\n",
    "# ∀A∃B(x ∈ A ⇔ x ∈ B ∧ P(x))"
   ]
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       "{2, 4}"
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   "source": [
    "A = {1, 2, 3, 4}\n",
    "B = {1, 2, 3, 4}\n",
    "\n",
    "{x for x in B if x % 2 == 0 and x in A and x in B} # forma um novo conjunto baseado na condição p(x) -> x mod 2 == 0"
   ]
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   "source": [
    "### d) Ax. do par\n",
    "\n",
    "\n",
    "# ∀A∀B∃C(x ∈ C ⇔ x ∈ A ∨ x ∈ B)"
   ]
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     "text": [
      "[{1, 2, 3, 4}, {8, 5, 6, 7}]\n"
     ]
    }
   ],
   "source": [
    "A = {1 ,2, 3, 4}\n",
    "B = {5, 6, 7, 8}\n",
    "\n",
    "C = [{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}]\n",
    "\n",
    "print([x for x in C if x == A or x == B])"
   ]
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   "source": [
    "### e) Ax. da união\n",
    "\n",
    "\n",
    "# ∀A∀B∃C∀x(x ∈ C ⇔ x ∈ A ∨ x ∈ B)"
   ]
  },
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     "text": [
      "{1, 2, 3, 4, 5, 6, 7, 8}\n"
     ]
    }
   ],
   "source": [
    "A = {1 ,2, 3, 4}\n",
    "B = {5, 6, 7, 8}\n",
    "\n",
    "C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}\n",
    "\n",
    "print({x for x in C if x in A or x in B})"
   ]
  },
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   "source": [
    "### f) Ax. da união generalizada\n",
    "\n",
    "# ∀x[x ∈ B ⇔ ∃b ∈ A (x ∈ B)]"
   ]
  },
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   "source": [
    "### g) Ax. do subconjunto\n",
    "\n",
    "# ∀A∀B∀x(x ∈ B ⇔ x ∈ A ∧ P(x)]"
   ]
  },
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   "source": [
    "### h) Ax. power set\n",
    "\n",
    "# ∀A∃B∀x(x ∈ B ⇔ x ⊆ A)"
   ]
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