(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 60035, 1854] NotebookOptionsPosition[ 54876, 1702] NotebookOutlinePosition[ 55264, 1719] CellTagsIndexPosition[ 55221, 1716] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Introduction.", "Section", CellChangeTimes->{{3.42324793675*^9, 3.423247942140625*^9}}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has many useful built-in functions. They are capitalized and use square \ brackets. For example, let's try Prime." }], "Text", CellChangeTimes->{{3.423314480640625*^9, 3.423314546515625*^9}}], Cell["We can use the Table function to list prime numbers.", "Text", CellChangeTimes->{{3.423314901796875*^9, 3.423314902046875*^9}, { 3.42331497128125*^9, 3.42331498978125*^9}}], Cell[TextData[{ "Exercise. Is ", Cell[BoxData[ FormBox["123456789", TraditionalForm]]], " prime." }], "Text", CellChangeTimes->{{3.423319726953125*^9, 3.423319755796875*^9}}], Cell["We can factor a number using FactorInteger", "Text", CellChangeTimes->{{3.42331523159375*^9, 3.423315244234375*^9}}], Cell["Use PrimeQ to determine whether a number is prime.", "Text", CellChangeTimes->{{3.423315321375*^9, 3.423315339875*^9}}], Cell["Exercise: Find a 9 digit prime number.", "Text", CellChangeTimes->{{3.42331546128125*^9, 3.423315475765625*^9}}], Cell[TextData[{ "Floor[x] is the greatest integer ", Cell[BoxData[ FormBox["\[LessEqual]", TraditionalForm]]], " x, and Ceiling[x] is the smallest integer ", Cell[BoxData[ FormBox["\[GreaterEqual]", TraditionalForm]]], " x. " }], "Text", CellChangeTimes->{{3.42331982246875*^9, 3.423319894609375*^9}, { 3.423350819703125*^9, 3.423350819765625*^9}}], Cell[TextData[{ "Exercise: How many multiples of 47 are there between ", Cell[BoxData[ FormBox[ RowBox[{"1000000", " ", "and", " ", "1000000000000000000"}], TraditionalForm]]], "?" }], "Text", CellChangeTimes->{{3.423320091484375*^9, 3.423320148671875*^9}, { 3.423350977796875*^9, 3.423351003953125*^9}, {3.423351078453125*^9, 3.42335110190625*^9}}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has a numer of built-in real numbers that can be displayed with arbitrary \ precision." }], "Text", CellChangeTimes->{{3.423320644078125*^9, 3.423320690265625*^9}}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " nicely handles matrices. We can define a matrix as a list of lists. Let's \ define the matrix ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"1", "2"}, {"3", "4"} }], ")"}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.423247965796875*^9, 3.42324806565625*^9}, { 3.42331441328125*^9, 3.42331441596875*^9}, {3.423314451625*^9, 3.4233144623125*^9}}], Cell["We can also use the pallette to define the matrix.", "Text", CellChangeTimes->{{3.42324817696875*^9, 3.423248200484375*^9}}], Cell["As well we can use the Insert menu.", "Text", CellChangeTimes->{{3.423248217609375*^9, 3.423248231640625*^9}}], Cell["\<\ To display the matrix as an array we use the MatrixForm function.\ \>", "Text", CellChangeTimes->{{3.42324824271875*^9, 3.423248273453125*^9}}], Cell["We can add, multiply, and scalar multiply matrices.", "Text", CellChangeTimes->{{3.423248479859375*^9, 3.423248515921875*^9}}], Cell["Using MatrixPower we can raise matrices to various powers.", "Text", CellChangeTimes->{{3.42324868140625*^9, 3.42324871309375*^9}}], Cell["\<\ If the entries of a matrix have a nice formula then it is useful to use the \ Table function.\ \>", "Text", CellChangeTimes->{{3.42324906378125*^9, 3.4232491024375*^9}}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can be used to make user defined functions. For example we can define a \ matrix that corresponds with pascal's triangle. This is an example of \ functional programming." }], "Text", CellChangeTimes->{{3.423250320609375*^9, 3.4232503543125*^9}, { 3.423320884921875*^9, 3.423320928578125*^9}, {3.423321706*^9, 3.423321722609375*^9}}], Cell["\<\ Let's write a program that computes the row sums of a matrix.\ \>", "Text", CellChangeTimes->{{3.423321727828125*^9, 3.42332176859375*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"rowsumlist", "[", "m_", "]"}], ":=", " ", RowBox[{"Table", "[", RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"j", "=", "1"}], RowBox[{"Length", "[", RowBox[{"m", "[", RowBox[{"[", "1", "]"}], "]"}], "]"}]], RowBox[{"m", "[", RowBox[{"[", RowBox[{"i", ",", "j"}], "]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"Length", "[", "m", "]"}]}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.42332177271875*^9, 3.423321899859375*^9}, { 3.423321935265625*^9, 3.423321948*^9}}], Cell["\<\ Exercise: What can you say about the row sums of pascal's triangle?\ \>", "Text", CellChangeTimes->{{3.4233220191875*^9, 3.423322051765625*^9}}], Cell["\<\ We can write programs to perform basic row operations on a matrix. The \ program rowadd uses matrix m and adds r times row i to row j.\ \>", "Text", CellChangeTimes->{{3.423248748984375*^9, 3.42324879134375*^9}, { 3.42325017678125*^9, 3.42325021965625*^9}, {3.423322174046875*^9, 3.423322188421875*^9}, 3.42332277375*^9, {3.423351484046875*^9, 3.42335149271875*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"rowadd", "[", RowBox[{"m_", ",", "r_", ",", "i_", ",", "j_"}], "]"}], ":=", " ", RowBox[{"Table", "[", RowBox[{ RowBox[{"If", "[", RowBox[{ RowBox[{"k", "\[Equal]", "j"}], ",", RowBox[{ RowBox[{"r", " ", RowBox[{"m", "[", RowBox[{"[", "i", "]"}], "]"}]}], "+", RowBox[{"m", "[", RowBox[{"[", "j", "]"}], "]"}]}], ",", RowBox[{"m", "[", RowBox[{"[", "k", "]"}], "]"}]}], "]"}], ",", RowBox[{"{", RowBox[{"k", ",", RowBox[{"Length", "[", "m", "]"}]}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.423250580953125*^9, 3.42325065878125*^9}}], Cell[TextData[{ "We can use row operations to find the inverse of ", Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0"}, {"1", "1", "0"}, {"1", "2", "1"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], CellChangeTimes->{3.423322271453125*^9, 3.423322398765625*^9}], "." }], "Text", CellChangeTimes->{{3.423322864828125*^9, 3.423322903359375*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "1", "0", "0"}, {"1", "1", "0", "0", "1", "0"}, {"1", "2", "1", "0", "0", "1"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Input"], Cell["\<\ Exercise: What can we say in general about the inverse of the pascal matrix?\ \>", "Text", CellChangeTimes->{{3.423351523453125*^9, 3.4233515530625*^9}}], Cell["The function Det finds the determinant of a matrix.", "Text", CellChangeTimes->{{3.423251640765625*^9, 3.423251661921875*^9}}], Cell[TextData[{ "Exercise: Compute ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"det", "(", "A", ")"}], RowBox[{"det", "(", SuperscriptBox["A", RowBox[{"-", "1"}]], ")"}]}], TraditionalForm]]], " for several examples of square matrices. What do you see?" }], "Text", CellChangeTimes->{{3.423251880015625*^9, 3.423251883125*^9}, { 3.423251915265625*^9, 3.423252034390625*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Cayley-Hamilton Theorem.", "Section", CellChangeTimes->{{3.4232521989375*^9, 3.42325222625*^9}}], Cell["\<\ We can use the IdentityMatrix function to define identity matrices.\ \>", "Text", CellChangeTimes->{{3.42325236665625*^9, 3.423252374265625*^9}, { 3.423252421109375*^9, 3.423252462921875*^9}}], Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"IdentityMatrix", "[", "4", "]"}], "]"}]], "Input", CellChangeTimes->{{3.42356175203125*^9, 3.423561766*^9}}], Cell[TextData[{ "Exercise: For several examples of square matrices, ", Cell[BoxData[ FormBox["A", TraditionalForm]]], ", find ", Cell[BoxData[ FormBox[ RowBox[{"det", "(", RowBox[{ RowBox[{"x", " ", "I"}], "-", "A"}], ")"}], TraditionalForm]]], ". What can you say in general? Let m1 be 2 by 2 and m2 be 3 by 3." }], "Text", CellChangeTimes->{{3.423252255296875*^9, 3.4232523399375*^9}, { 3.42356208721875*^9, 3.423562111015625*^9}}], Cell[BoxData[ RowBox[{"Det", "[", RowBox[{ RowBox[{"x", " ", RowBox[{"IdentityMatrix", "[", "2", "]"}]}], "-", "m1"}], "]"}]], "Input",\ CellChangeTimes->{{3.423561822953125*^9, 3.42356184853125*^9}, 3.42356188065625*^9}], Cell[BoxData[ RowBox[{"Det", "[", RowBox[{ RowBox[{"x", " ", RowBox[{"IdentityMatrix", "[", "3", "]"}]}], "-", "m2"}], "]"}]], "Input",\ CellChangeTimes->{{3.423561822953125*^9, 3.42356184853125*^9}, 3.42356188065625*^9, {3.423561951453125*^9, 3.4235619540625*^9}}], Cell[TextData[{ "The polynomial ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"p", "(", "x", ")"}], "=", RowBox[{"det", "(", RowBox[{ RowBox[{"x", " ", "I"}], "-", "A"}], ")"}]}], TraditionalForm]]], " is called the characteristic polynomial of the square matrix ", Cell[BoxData[ FormBox["A", TraditionalForm]]], ". " }], "Text", CellChangeTimes->{{3.423323090328125*^9, 3.423323165734375*^9}}], Cell[TextData[{ "Exercise: If the characteristic polynomial of ", Cell[BoxData[ FormBox["A", TraditionalForm]]], " has the form ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"p", "(", "x", ")"}], "=", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["a", "0"], "+", RowBox[{ SubscriptBox["a", "1"], "x"}], "+"}], "..."}], "+", RowBox[{ SubscriptBox["a", "n"], SuperscriptBox["x", "n"]}]}]}], TraditionalForm]]], " what can you say about the matrix ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"p", "(", "A", ")"}], "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["a", "0"], "I"}], "+", RowBox[{ SubscriptBox["a", "1"], "A"}], "+"}], "..."}], "+", RowBox[{ SubscriptBox["a", "n"], SuperscriptBox["A", "n"]}]}]}], TraditionalForm]]], "?" }], "Text", CellChangeTimes->{{3.42332318571875*^9, 3.4233233933125*^9}}], Cell["\<\ We can use this to find the inverse of a matrix by taking powers.\ \>", "Text", CellChangeTimes->{{3.4233234133125*^9, 3.423323438546875*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Fibonacci sequence.", "Section", CellChangeTimes->{{3.423323503296875*^9, 3.4233235130625*^9}}], Cell["Here are the first few Fibonacci numbers.", "Text", CellChangeTimes->{{3.4233235548125*^9, 3.42332355725*^9}, { 3.423323600859375*^9, 3.42332361575*^9}}], Cell[TextData[{ "Let ", Cell[BoxData[ FormBox[ SubscriptBox["f", "n"], TraditionalForm]]], " be the ", Cell[BoxData[ FormBox[ SuperscriptBox["n", "th"], TraditionalForm]]], " Fibonacci number. Then ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["f", "0"], "=", "0"}], ",", " ", RowBox[{ SubscriptBox["f", "1"], "=", "1"}], ",", " ", RowBox[{ RowBox[{"and", " ", "for", " ", "n"}], "\[GreaterEqual]", "2"}], ",", " ", RowBox[{ SubscriptBox["f", "n"], "=", RowBox[{ SubscriptBox["f", RowBox[{"n", "-", "1"}]], "+", SubscriptBox["f", RowBox[{"n", "-", "2"}]]}]}]}], TraditionalForm]]], ". This last equation is called a finite difference equation for the \ sequence ", Cell[BoxData[ FormBox[ SubscriptBox["f", "n"], TraditionalForm]]], ". Recall that a geometric sequence has the form ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["r", "n"], " ", "for", " ", "n"}], "=", "0"}], ",", "1", ",", "2", ",", "..."}], TraditionalForm]]], ". ; we assume ", Cell[BoxData[ FormBox[ RowBox[{"r", "\[NotEqual]", "0"}], TraditionalForm]]], ". " }], "Text", CellChangeTimes->{{3.42332366571875*^9, 3.42332393925*^9}, { 3.42332400928125*^9, 3.42332404984375*^9}}], Cell[TextData[{ "Exercise: For which values of ", Cell[BoxData[ FormBox["r", TraditionalForm]]], " does the geometric sequence ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["r", "n"], " "}], TraditionalForm]]], "satisfy the same finite difference equation as does the Fibonacci \ sequence?" }], "Text", CellChangeTimes->{{3.423324081953125*^9, 3.423324158328125*^9}}], Cell[TextData[{ "There are some theoretical properties we need. \n1. Suppose ", Cell[BoxData[ FormBox[ SubscriptBox["a", "n"], TraditionalForm]]], " satisfies the same finite difference equation as does the Fibonacci \ sequence. The so does ", Cell[BoxData[ FormBox[ RowBox[{"c", " ", SubscriptBox["a", "n"]}], TraditionalForm]]], " for any constant ", Cell[BoxData[ FormBox["c", TraditionalForm]]], ".\n2. Suppose ", Cell[BoxData[ FormBox[ SubscriptBox["a", "n"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["b", "n"], TraditionalForm]]], " satisfy the same finite difference equation as does the Fibonacci \ sequence. The so does ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["a", "n"], "+", SubscriptBox["b", "n"]}], TraditionalForm]]], ".\n3. Any sequence that satisfies the same finite difference equation as \ does the Fibonacci sequence is completely determined by the ", Cell[BoxData[ FormBox[ SuperscriptBox["0", "th"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SuperscriptBox["1", "st"], TraditionalForm]]], " terms.\n\nLet ", Cell[BoxData[ FormBox[ RowBox[{"r", "=", FractionBox[ RowBox[{"1", "+", SqrtBox["5"]}], "2"]}], TraditionalForm]]], ". Then ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["f", "n"], "=", RowBox[{ RowBox[{"a", " ", SuperscriptBox["r", "n"]}], "+", SuperscriptBox[ RowBox[{"b", "(", FractionBox[ RowBox[{"-", "1"}], "r"], ")"}], "n"]}]}], TraditionalForm]]], "where ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"a", "+", "b"}], "=", RowBox[{ RowBox[{ RowBox[{"0", " ", "and", " ", "a", " ", "r"}], "-", FractionBox["b", "r"]}], "=", "1"}]}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.423324699359375*^9, 3.42332487775*^9}, { 3.42332492709375*^9, 3.42332502346875*^9}, {3.4233260916875*^9, 3.423326094125*^9}, {3.4233451116875*^9, 3.42334515621875*^9}, { 3.423345200046875*^9, 3.42334534884375*^9}, {3.4233453919375*^9, 3.423345405625*^9}, 3.423345437796875*^9}], Cell[BoxData[ RowBox[{"r", "=", FractionBox[ RowBox[{"1", "+", SqrtBox["5"]}], "2"]}]], "Input", CellChangeTimes->{{3.42334535321875*^9, 3.42334537009375*^9}}], Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"a", "+", "b"}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{ RowBox[{"a", " ", "r"}], "-", FractionBox["b", "r"]}], "\[Equal]", "1"}]}], "}"}], "]"}]], "Input", CellChangeTimes->{{3.42334537609375*^9, 3.423345451859375*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"a", "=", FractionBox[ RowBox[{"1", "+", SqrtBox["5"]}], RowBox[{"5", "+", SqrtBox["5"]}]]}], ";", RowBox[{"b", "=", RowBox[{"-", "a"}]}]}]], "Input", CellChangeTimes->{{3.42334553709375*^9, 3.42334556496875*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"g", "[", "n_", "]"}], ":=", " ", RowBox[{ RowBox[{"a", " ", SuperscriptBox["r", "n"]}], "+", RowBox[{"b", SuperscriptBox[ RowBox[{"(", FractionBox[ RowBox[{"-", "1"}], "r"], ")"}], "n"]}]}]}]], "Input", CellChangeTimes->{{3.423345690359375*^9, 3.423345731453125*^9}}], Cell[BoxData[ RowBox[{"FullSimplify", "[", RowBox[{"g", "[", "n", "]"}], "]"}]], "Input", CellChangeTimes->{{3.423345737421875*^9, 3.423345749953125*^9}}], Cell[BoxData[ RowBox[{"FullSimplify", "[", RowBox[{"{", RowBox[{ RowBox[{"Fibonacci", "[", "20", "]"}], ",", RowBox[{"g", "[", "20", "]"}]}], "}"}], "]"}]], "Input", CellChangeTimes->{{3.423345792375*^9, 3.423345863421875*^9}}], Cell[TextData[{ "Note that for large ", Cell[BoxData[ FormBox["n", TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{"Fibonacci", "[", "n", "]"}], TraditionalForm]]], " is approximately ", Cell[BoxData[ FormBox[ RowBox[{"a", " ", SuperscriptBox["r", "n"]}], TraditionalForm]]], ", i.e. the Fibonacci sequence is approximately geometric for large ", Cell[BoxData[ FormBox["n", TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.423345874125*^9, 3.423345877203125*^9}, { 3.423345927140625*^9, 3.4233460576875*^9}}], Cell[BoxData[ RowBox[{"TableForm", "[", RowBox[{"Table", "[", RowBox[{ RowBox[{"N", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Fibonacci", "[", "n", "]"}], ",", RowBox[{"a", " ", SuperscriptBox["r", "n"]}]}], "}"}], ",", "21"}], "]"}], ",", RowBox[{"{", RowBox[{"n", ",", "10", ",", "100", ",", "10"}], "}"}]}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.42334607065625*^9, 3.423346127265625*^9}, { 3.42334618559375*^9, 3.42334623971875*^9}, {3.42334630715625*^9, 3.423346307640625*^9}}], Cell[BoxData[ RowBox[{"Fibonacci", "[", "100", "]"}]], "Input", CellChangeTimes->{{3.4233462641875*^9, 3.42334627025*^9}}], Cell[BoxData[ RowBox[{"N", "[", RowBox[{ RowBox[{"a", " ", SuperscriptBox["r", "100"]}], ",", "21"}], "]"}]], "Input", CellChangeTimes->{{3.423346280296875*^9, 3.42334628828125*^9}, { 3.423346322875*^9, 3.42334633328125*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Inclusion/Exclusion.", "Section", CellChangeTimes->{{3.42334641015625*^9, 3.42334642234375*^9}}], Cell[TextData[{ "Suppose ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["A", "1"], ",", " ", SubscriptBox["A", "2"], ",", " ", RowBox[{"and", " ", SubscriptBox["A", "3"]}]}], TraditionalForm]]], " are finite subsets of a finite universal set ", Cell[BoxData[ FormBox["U", TraditionalForm]]], ". Let ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["S", "0"], "=", RowBox[{"n", "(", "U", ")"}]}], TraditionalForm]]], ", the number of elements in ", Cell[BoxData[ FormBox["U", TraditionalForm]]], " which is also called the cardinal number of ", Cell[BoxData[ FormBox["U", TraditionalForm]]], ". For ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"i", "=", "1"}], ",", "2", ",", " ", RowBox[{"and", " ", "3"}]}], TraditionalForm]]], " let ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["S", "i"], "=", RowBox[{"the", " ", "sum", " ", "of", " ", "the", " "}]}], TraditionalForm]]], "cardinal numbers of sets formed by intersecting ", Cell[BoxData[ FormBox["i", TraditionalForm]]], " of the sets of ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["A", "1"], ",", " ", SubscriptBox["A", "2"], ",", " ", SubscriptBox["A", "3"]}], TraditionalForm]]], ". Also, for ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"i", "=", "0"}], ",", "1", ",", "2", ",", "3"}], TraditionalForm]]], " let ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["N", "i"], "="}], TraditionalForm]]], " the number of elements found in exactly ", Cell[BoxData[ FormBox["i", TraditionalForm]]], " of the sets ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["A", "1"], ",", " ", SubscriptBox["A", "2"], ",", " ", SubscriptBox["A", "3"]}], TraditionalForm]]], ".\n\nClearly ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["S", "0"], "=", RowBox[{ SubscriptBox["N", "0"], "+", SubscriptBox["N", "1"], "+", SubscriptBox["N", "2"], "+", SubscriptBox["N", "3"]}]}], TraditionalForm]]], ".\n\nAlso, ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["S", "1"], "=", RowBox[{ RowBox[{ RowBox[{"n", "(", SubscriptBox["A", "1"], ")"}], "+", RowBox[{"n", "(", SubscriptBox["A", "2"], ")"}], "+", RowBox[{"n", "(", SubscriptBox["A", "3"], ")"}]}], "=", RowBox[{ SubscriptBox["N", "1"], "+", RowBox[{"2", 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