34.2-10

Chapter_34_NP-Completeness/exercises_34.2.ipynb

@@ -227,6 +227,28 @@
     "\n",
     "$\\text{P} \\subseteq \\text{co-NP}$"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### 34.2-10\n",
+    "\n",
+    "> Prove that if $\\text{NP} \\ne \\text{co-NP}$, then $\\text{P} \\ne \\text{NP}$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Prove the contrapositive.\n",
+    "\n",
+    "Suppose $\\text{P} = \\text{NP}$:\n",
+    "\n",
+    "* $\\forall L \\in \\text{NP}$ $\\Rightarrow$ $L \\in \\text{P}$ $\\Rightarrow$ $\\overline{L} \\in \\text{P}$ $\\Rightarrow$ $L \\in \\text{co-NP}$\n",
+    "* $\\forall L \\in \\text{co-NP}$ $\\Rightarrow$ $\\overline{L} \\in \\text{NP}$ $\\Rightarrow$ $\\overline{L} \\in \\text{P}$ $\\Rightarrow$ $L \\in \\text{P}$\n",
+    "*  $\\Rightarrow$ $\\text{NP} = \\text{co-NP}$."
+   ]
   }
  ],
  "metadata": {