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    "# Question 42\n",
    "## c.\n",
    "$$S \\rightarrow 0S0 | 0S1 | 1S0 | 1S1 | 1$$\n",
    "\n",
    "## d.\n",
    "$$S \\rightarrow X | \\epsilon$$\n",
    "$$X \\rightarrow aXdd | Y$$\n",
    "$$Y \\rightarrow bYd | Z$$\n",
    "$$Z \\rightarrow cZ | \\epsilon$$\n",
    "\n",
    "## e.\n",
    "$$S \\rightarrow 0S | 1S | 00S | 01S | 10S | 11S | 000S | 001S | 010S | 011S| 100S | 101S | 111S | \\epsilon$$"
   ]
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    "$\\pagebreak$"
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   "source": [
    "# Question 43\n",
    "## a.\n",
    "$$B = \\{ w \\in \\{a, b, c\\}^* | |w| = 4n+1, n \\in \\mathbb{Z}^{\\text{nonneg}} \\}$$\n",
    "\n",
    "## b.\n",
    "$$B = \\{ 0^2x1^3y+2 | x, y \\in  \\mathbb{Z}^{\\text{nonneg}} \\}$$\n",
    "\n",
    "## c.\n",
    "$$B = \\{ 0^x1^y | x \\neq y \\text{ and } x, y \\in  \\mathbb{Z}^{\\text{nonneg}} \\}$$"
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    "$\\pagebreak$"
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   "source": [
    "# Question 44\n",
    "## a.\n",
    "$$B = \\{ 0^x1^y | y = x + 1 \\text{ and } x, y \\in  \\mathbb{Z}^{\\text{nonneg}} \\}$$\n",
    "\n",
    "## c.\n",
    "$$B = \\{ 0^x1^y | y = 3x \\text{ and } x, y \\in  \\mathbb{Z}^{\\text{nonneg}} \\}$$"
   ]
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    "$\\pagebreak$"
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   "metadata": {},
   "source": [
    "# Question 45\n",
    "1. First the machine should move left to right looking for a \"b\". If it does not find one it moves to the reject state and if it finds one it will:\n",
    "   - Erase the \"b\"\n",
    "   - Move back to the start\n",
    "   - Look for an \"a\". If it finds one it will repeat step 1. If it does not find one, it will move to the accept state"
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   "source": [
    "# Question 47\n",
    "1. First the machine should check if the length of the string is at least 6 and if not, reject\n",
    "2. Copy 0 or more symbols to tape 2 from tape 1\n",
    "3. Erase all of tape 2\n",
    "4. Copy 3 symbols to tape 2 from tape 1\n",
    "5. Copy 0 or more symbols to tape 3 from tape 1\n",
    "6. Erase of all of tape 3\n",
    "7. Copy 3 symbols to tape 3 from tape 1\n",
    "8. Compare tapes 2 and 3 and if they match, accept, otherwise reject\n",
    "\n",
    "The nondeterminism of this machine is the copying 0 or more symbols, as this accounts for strings u and x, as they can be any length including 0, and the machine can make a choice of 0 or more."
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