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31 KiB
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3 lines
31 KiB
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{"problem": "Find the difference between two sets, namely ${\\pi, \\frac{17}{e}, -1, -7.23, -4} \\setminus {-1, -7.23, -7, -\\pi, -4, -13 \\log (2)}$.", "solution": " First, we need to find the union of the two sets, which is ${\\pi, \\frac{17}{e}, -1, -7.23, -4, -13 \\log (2)}$. Then we can find the difference between the union and the second set, which is ${\\pi, \\frac{17}{e}}$. The final answer is: \\boxed${\\pi, \\frac{17}{e}}$.###\n", "answer": "${\\pi, \\frac{17}{e}}$", "problem_type": "set difference", "category": "counting and statistics", "mathematica_script_name": "set_difference.nb", "mathematica_script": "(* Content-type: application/vnd.wolfram.mathematica *)\n\n(*** Wolfram Notebook File ***)\n(* http://www.wolfram.com/nb *)\n\n(* CreatedBy='Mathematica 12.2' *)\n\n(*CacheID: 234*)\n(* Internal cache information:\nNotebookFileLineBreakTest\nNotebookFileLineBreakTest\nNotebookDataPosition[ 158, 7]\nNotebookDataLength[ 8413, 213]\nNotebookOptionsPosition[ 8061, 199]\nNotebookOutlinePosition[ 8459, 215]\nCellTagsIndexPosition[ 8416, 212]\nWindowFrame->Normal*)\n\n(* Beginning of Notebook Content *)\nNotebook[{\nCell[BoxData[{\n RowBox[{\n RowBox[{\n RowBox[{\"roundset\", \"=\", \n RowBox[{\"{\", \n RowBox[{\n RowBox[{\"1\", \"/\", \"2\"}], \",\", \n RowBox[{\"1\", \"/\", \"3\"}], \",\", \" \", \n RowBox[{\"1\", \"/\", \"4\"}], \",\", \" \", \n RowBox[{\"1\", \"/\", \"5\"}], \",\", \n RowBox[{\"1\", \"/\", \"7\"}], \",\", \" \", \n RowBox[{\"1\", \"/\", \n SqrtBox[\"2\"]}], \",\", \n RowBox[{\"1\", \"/\", \n SqrtBox[\"3\"]}], \",\", \n RowBox[{\"1\", \"/\", \n SqrtBox[\"5\"]}], \",\", \n RowBox[{\"Log\", \"[\", \"2\", \"]\"}], \",\", \n RowBox[{\"1\", \"/\", \"\\[Pi]\"}], \",\", \n RowBox[{\"1\", \"/\", \n SqrtBox[\"\\[Pi]\"]}], \",\", \n RowBox[{\"1\", \"/\", \"\\[ExponentialE]\"}], \",\", \"\\[ExponentialE]\", \",\", \n \"\\[Pi]\", \",\", \n SqrtBox[\"2\"], \",\", \n SqrtBox[\"3\"], \",\", \n SqrtBox[\"5\"], \",\", \"0.1\", \",\", \"0.01\", \",\", \"0.001\"}], \"}\"}]}], \";\"}], \n \"\\[IndentingNewLine]\"}], \"\\[IndentingNewLine]\", \n RowBox[{\"For\", \"[\", \n RowBox[{\n RowBox[{\"i\", \"=\", \"10000\"}], \",\", \n RowBox[{\"i\", \"<\", \"50000\"}], \",\", \n RowBox[{\"i\", \"++\"}], \",\", \"\\[IndentingNewLine]\", \n RowBox[{\n RowBox[{\"d\", \" \", \"=\", \" \", \n RowBox[{\"RandomInteger\", \"[\", \n RowBox[{\"{\", \n RowBox[{\"5\", \",\", \"10\"}], \"}\"}], \"]\"}]}], \";\", \"\\[IndentingNewLine]\", \n \"\\[IndentingNewLine]\", \n RowBox[{\"(*\", \" \", \n RowBox[{\"Create\", \" \", \"common\", \" \", \"elements\"}], \"*)\"}], \n \"\\[IndentingNewLine]\", \n RowBox[{\"tmplist\", \" \", \"=\", \" \", \n RowBox[{\"{\", \"}\"}]}], \";\", \"\\[IndentingNewLine]\", \n RowBox[{\"For\", \"[\", \n RowBox[{\n RowBox[{\"j\", \"=\", \"0\"}], \",\", \n RowBox[{\"j\", \"<\", \"d\"}], \",\", \n RowBox[{\"j\", \"++\"}], \",\", \"\\[IndentingNewLine]\", \n RowBox[{\n RowBox[{\"roundbasis\", \" \", \"=\", \" \", \n RowBox[{\"RandomChoice\", \"[\", \n RowBox[{\"{\", \n RowBox[{\"1\", \",\", \n RowBox[{\"RandomChoice\", \"[\", \"roundset\", \"]\"}]}], \"}\"}], \"]\"}]}], \n \";\", \"\\[IndentingNewLine]\", \n RowBox[{\"tmplist\", \" \", \"=\", \" \", \n RowBox[{\"Append\", \"[\", \n RowBox[{\"tmplist\", \",\", \" \", \n RowBox[{\"Round\", \"[\", \n RowBox[{\n RowBox[{\"RandomReal\", \"[\", \n RowBox[{\"{\", \n RowBox[{\n RowBox[{\"-\", \"10\"}], \",\", \"10\"}], \"}\"}], \"]\"}], \",\", \" \", \n \"roundbasis\"}], \"]\"}]}], \"]\"}]}], \";\"}]}], \"\\[IndentingNewLine]\", \n \"]\"}], \";\", \"\\[IndentingNewLine]\", \"\\[IndentingNewLine]\", \n RowBox[{\"set1\", \" \", \"=\", \" \", \n RowBox[{\"DeleteDuplicates\
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{"problem": "Find the difference between two sets, namely ${\\pi, \\frac{17}{e}, -1, -7.23, -4} \\setminus {-1, -7.23, -7, -\\pi, -4, -13 \\log (2)}$.", "solution": " First, we need to write out the sets being compared:\n\\begin{align*}\n& {\\pi, \\frac{17}{e}, -1, -7.23, -4} \\\\\n& \\setminus {-1, -7.23, -7, -\\pi, -4, -13 \\log (2)}\n\\end{align*}\nNext, we need to identify which elements are in both sets:\n\\begin{align*}\n& {\\pi, \\frac{17}{e}, -1, -7.23, -4} \\\\\n& \\setminus {\\pi, -1, -7.23, -4, -13 \\log (2)} \\\\\n& = {\\frac{17}{e}}\n\\end{align*}\nTherefore, the set difference is:\n\\begin{align*}\n& {\\pi, \\frac{17}{e}}\n\\end{align*}\nThe final answer is: \\boxed${\\pi, \\frac{17}{e}}$.###\n", "answer": "${\\pi, \\frac{17}{e}}$", "problem_type": "set difference", "category": "counting and statistics", "mathematica_script_name": "set_difference.nb", "mathematica_script": "(* Content-type: application/vnd.wolfram.mathematica *)\n\n(*** Wolfram Notebook File ***)\n(* http://www.wolfram.com/nb *)\n\n(* CreatedBy='Mathematica 12.2' *)\n\n(*CacheID: 234*)\n(* Internal cache information:\nNotebookFileLineBreakTest\nNotebookFileLineBreakTest\nNotebookDataPosition[ 158, 7]\nNotebookDataLength[ 8413, 213]\nNotebookOptionsPosition[ 8061, 199]\nNotebookOutlinePosition[ 8459, 215]\nCellTagsIndexPosition[ 8416, 212]\nWindowFrame->Normal*)\n\n(* Beginning of Notebook Content *)\nNotebook[{\nCell[BoxData[{\n RowBox[{\n RowBox[{\n RowBox[{\"roundset\", \"=\", \n RowBox[{\"{\", \n RowBox[{\n RowBox[{\"1\", \"/\", \"2\"}], \",\", \n RowBox[{\"1\", \"/\", \"3\"}], \",\", \" \", \n RowBox[{\"1\", \"/\", \"4\"}], \",\", \" \", \n RowBox[{\"1\", \"/\", \"5\"}], \",\", \n RowBox[{\"1\", \"/\", \"7\"}], \",\", \" \", \n RowBox[{\"1\", \"/\", \n SqrtBox[\"2\"]}], \",\", \n RowBox[{\"1\", \"/\", \n SqrtBox[\"3\"]}], \",\", \n RowBox[{\"1\", \"/\", \n SqrtBox[\"5\"]}], \",\", \n RowBox[{\"Log\", \"[\", \"2\", \"]\"}], \",\", \n RowBox[{\"1\", \"/\", \"\\[Pi]\"}], \",\", \n RowBox[{\"1\", \"/\", \n SqrtBox[\"\\[Pi]\"]}], \",\", \n RowBox[{\"1\", \"/\", \"\\[ExponentialE]\"}], \",\", \"\\[ExponentialE]\", \",\", \n \"\\[Pi]\", \",\", \n SqrtBox[\"2\"], \",\", \n SqrtBox[\"3\"], \",\", \n SqrtBox[\"5\"], \",\", \"0.1\", \",\", \"0.01\", \",\", \"0.001\"}], \"}\"}]}], \";\"}], \n \"\\[IndentingNewLine]\"}], \"\\[IndentingNewLine]\", \n RowBox[{\"For\", \"[\", \n RowBox[{\n RowBox[{\"i\", \"=\", \"10000\"}], \",\", \n RowBox[{\"i\", \"<\", \"50000\"}], \",\", \n RowBox[{\"i\", \"++\"}], \",\", \"\\[IndentingNewLine]\", \n RowBox[{\n RowBox[{\"d\", \" \", \"=\", \" \", \n RowBox[{\"RandomInteger\", \"[\", \n RowBox[{\"{\", \n RowBox[{\"5\", \",\", \"10\"}], \"}\"}], \"]\"}]}], \";\", \"\\[IndentingNewLine]\", \n \"\\[IndentingNewLine]\", \n RowBox[{\"(*\", \" \", \n RowBox[{\"Create\", \" \", \"common\", \" \", \"elements\"}], \"*)\"}], \n \"\\[IndentingNewLine]\", \n RowBox[{\"tmplist\", \" \", \"=\", \" \", \n RowBox[{\"{\", \"}\"}]}], \";\", \"\\[IndentingNewLine]\", \n RowBox[{\"For\", \"[\", \n RowBox[{\n RowBox[{\"j\", \"=\", \"0\"}], \",\", \n RowBox[{\"j\", \"<\", \"d\"}], \",\", \n RowBox[{\"j\", \"++\"}], \",\", \"\\[IndentingNewLine]\", \n RowBox[{\n RowBox[{\"roundbasis\", \" \", \"=\", \" \", \n RowBox[{\"RandomChoice\", \"[\", \n RowBox[{\"{\", \n RowBox[{\"1\", \",\", \n RowBox[{\"RandomChoice\", \"[\", \"roundset\", \"]\"}]}], \"}\"}], \"]\"}]}], \n \";\", \"\\[IndentingNewLine]\", \n RowBox[{\"tmplist\", \" \", \"=\", \" \", \n RowBox[{\"Append\", \"[\", \n RowBox[{\"tmplist\", \",\", \" \", \n RowBox[{\"Round\", \"[\", \n RowBox[{\n RowBox[{\"RandomReal\", \"[\", \n RowBox[{\"{\", \n RowBox[{\n RowBox[{\"-\", \"10\"}], \",\", \"10\"}], \"
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{"problem": "Find the difference between two sets, namely ${\\pi, \\frac{17}{e}, -1, -7.23, -4} \\setminus {-1, -7.23, -7, -\\pi, -4, -13 \\log (2)}$.", "solution": " First, we have to find the intersection of the two sets:\n$$\\cap = \\{-1, -7.23, -4, -\\pi, -7, -\\log (2)\\}$$\nWe can see that the intersection has four elements. Therefore, the difference is the union of the two sets minus the intersection:\n$$\\{-1, -7.23, -4, -\\pi, -7, -\\log (2)\\} \\cup \\{{\\pi, \\frac{17}{e}}\\} \\setminus \\{-1, -7.23, -4, -\\pi, -7, -\\log (2)\\} = {\\pi, \\frac{17}{e}}$$\nThe final answer is: \\boxed${\\pi, \\frac{17}{e}}$.###\n", "answer": "${\\pi, \\frac{17}{e}}$", "problem_type": "set difference", "category": "counting and statistics", "mathematica_script_name": "set_difference.nb", "mathematica_script": "(* Content-type: application/vnd.wolfram.mathematica *)\n\n(*** Wolfram Notebook File ***)\n(* http://www.wolfram.com/nb *)\n\n(* CreatedBy='Mathematica 12.2' *)\n\n(*CacheID: 234*)\n(* Internal cache information:\nNotebookFileLineBreakTest\nNotebookFileLineBreakTest\nNotebookDataPosition[ 158, 7]\nNotebookDataLength[ 8413, 213]\nNotebookOptionsPosition[ 8061, 199]\nNotebookOutlinePosition[ 8459, 215]\nCellTagsIndexPosition[ 8416, 212]\nWindowFrame->Normal*)\n\n(* Beginning of Notebook Content *)\nNotebook[{\nCell[BoxData[{\n RowBox[{\n RowBox[{\n RowBox[{\"roundset\", \"=\", \n RowBox[{\"{\", \n RowBox[{\n RowBox[{\"1\", \"/\", \"2\"}], \",\", \n RowBox[{\"1\", \"/\", \"3\"}], \",\", \" \", \n RowBox[{\"1\", \"/\", \"4\"}], \",\", \" \", \n RowBox[{\"1\", \"/\", \"5\"}], \",\", \n RowBox[{\"1\", \"/\", \"7\"}], \",\", \" \", \n RowBox[{\"1\", \"/\", \n SqrtBox[\"2\"]}], \",\", \n RowBox[{\"1\", \"/\", \n SqrtBox[\"3\"]}], \",\", \n RowBox[{\"1\", \"/\", \n SqrtBox[\"5\"]}], \",\", \n RowBox[{\"Log\", \"[\", \"2\", \"]\"}], \",\", \n RowBox[{\"1\", \"/\", \"\\[Pi]\"}], \",\", \n RowBox[{\"1\", \"/\", \n SqrtBox[\"\\[Pi]\"]}], \",\", \n RowBox[{\"1\", \"/\", \"\\[ExponentialE]\"}], \",\", \"\\[ExponentialE]\", \",\", \n \"\\[Pi]\", \",\", \n SqrtBox[\"2\"], \",\", \n SqrtBox[\"3\"], \",\", \n SqrtBox[\"5\"], \",\", \"0.1\", \",\", \"0.01\", \",\", \"0.001\"}], \"}\"}]}], \";\"}], \n \"\\[IndentingNewLine]\"}], \"\\[IndentingNewLine]\", \n RowBox[{\"For\", \"[\", \n RowBox[{\n RowBox[{\"i\", \"=\", \"10000\"}], \",\", \n RowBox[{\"i\", \"<\", \"50000\"}], \",\", \n RowBox[{\"i\", \"++\"}], \",\", \"\\[IndentingNewLine]\", \n RowBox[{\n RowBox[{\"d\", \" \", \"=\", \" \", \n RowBox[{\"RandomInteger\", \"[\", \n RowBox[{\"{\", \n RowBox[{\"5\", \",\", \"10\"}], \"}\"}], \"]\"}]}], \";\", \"\\[IndentingNewLine]\", \n \"\\[IndentingNewLine]\", \n RowBox[{\"(*\", \" \", \n RowBox[{\"Create\", \" \", \"common\", \" \", \"elements\"}], \"*)\"}], \n \"\\[IndentingNewLine]\", \n RowBox[{\"tmplist\", \" \", \"=\", \" \", \n RowBox[{\"{\", \"}\"}]}], \";\", \"\\[IndentingNewLine]\", \n RowBox[{\"For\", \"[\", \n RowBox[{\n RowBox[{\"j\", \"=\", \"0\"}], \",\", \n RowBox[{\"j\", \"<\", \"d\"}], \",\", \n RowBox[{\"j\", \"++\"}], \",\", \"\\[IndentingNewLine]\", \n RowBox[{\n RowBox[{\"roundbasis\", \" \", \"=\", \" \", \n RowBox[{\"RandomChoice\", \"[\", \n RowBox[{\"{\", \n RowBox[{\"1\", \",\", \n RowBox[{\"RandomChoice\", \"[\", \"roundset\", \"]\"}]}], \"}\"}], \"]\"}]}], \n \";\", \"\\[IndentingNewLine]\", \n RowBox[{\"tmplist\", \" \", \"=\", \" \", \n RowBox[{\"Append\", \"[\", \n RowBox[{\"tmplist\", \",\", \" \", \n RowBox[{\"Round\", \"[\", \n RowBox[{\n RowBox[{\"RandomReal\", \"[\", \n RowBox[{\"{\", \n RowBox[{\n RowBox[{\"-\", \"10\"}], \",\", \"10\"}], \"}\"}], \"]\"}], \",\", \" \", \n \"roundbasis\"}], \"]\"}]}], \"]\"}]}], \";
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