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Logika", "Section"], Cell[CellGroupData[{ Cell["\<\ P\:0159. 1.1 Zjednodu\[SHacek]te v\[YAcute]rok p \[And] (p \[Or] q)\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(LogicalExpand[p\ && \ \((p\ || q)\)]\)], "Input"], Cell[BoxData[ \(p\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ P\:0159. 1.2 Zjednodu\[SHacek]te v\[YAcute]rok \[Not](p \[Or] q) \ \[DoubleLeftRightArrow] \[Not]p \[And] \[Not]q\ \>", "Subsubsection"], Cell["\<\ Jedn\[AAcute] se vlastn\:011b o de Morgan\:016fv z\[AAcute]kon, tedy by m\ \:011blo vyj\[IAcute]t \"true\"\ \>", "Text"], Cell["\<\ Ekvivalence nen\[IAcute] v Mathematice definov\[AAcute]na, mus\[IAcute]me si \ pomoci jinak \ \>", "Text"], Cell["\<\ 1. pomoc\[IAcute] p\:0159episu ekvivalence do dvou implikac\[IAcute]:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(LogicalExpand[ Implies[\(! \((p || q)\)\), \(! p\)\ && \ \(! q\)] && Implies[\(! p\)\ && \ \(! q\), \(! \((p || q)\)\)]]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell["\<\ 2. definujeme si ekvivalenci jako funkci dvou prom\:011bnn\[YAcute]ch, funkci \ pak pou\:017eijeme k \:0159e\[SHacek]en\[IAcute] \[UAcute]lohy:\ \>", "Text"], Cell[BoxData[ \(Equivalence[a_, b_] := Implies[a, b] && Implies[b, a]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Equivalence[\(! \((p || q)\)\), \(! p\) && \(! q\)]\ // LogicalExpand\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ P\:0159. 1.3 Ur\[CHacek]ete, pro kter\[EAcute] hodnoty v\[YAcute]rok\:016f \ a, b, c je n\[AAcute]sleduj\[IAcute]c\[IAcute] v\[YAcute]rok nepravdiv\ \[YAcute]. ((\[Not]A\[DoubleRightArrow]C)\[Or](B\[And]\[Not]C))\ \[DoubleLeftRightArrow](A\[Or]C)\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(LogicalExpand[\(! Equivalence[\((Implies[\(! a\), c] || b && \ \(! c\))\), a || c]\)]\)], "Input"], Cell[BoxData[ \(b && \(! a\) && \(! c\)\)], "Output"] }, Open ]], Cell["\<\ Tedy v\[YAcute]rok je nepravdiv\[YAcute] pro jedinou kombinaci v\[YAcute]rok\ \:016f: a neplat\[IAcute], b plat\[IAcute], c neplat\[IAcute].\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ P\:0159. 1.4 Rozhodn\:011bte, zda v\[YAcute]rokov\[AAcute] formule je \ tautologie: (A \[And] B) \[DoubleRightArrow] (A \[Or] B)\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(LogicalExpand[Implies[A && B, A || B]]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["2. 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P\:0159irozen\[AAcute] \[CHacek]\[IAcute]sla", "Section"], Cell[CellGroupData[{ Cell["\<\ P\:0159. 3.1 Napi\[SHacek]te tabulku prvn\[IAcute]ch 100 prvo\[CHacek]\ \[IAcute]sel\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[Prime[n], {n, 100}]\)], "Input"], Cell[BoxData[ \({2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ P\:0159. 3.2 Vypi\[SHacek]te 2006. prvo\[CHacek]\[IAcute]slo\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Prime[2006]\)], "Input"], Cell[BoxData[ \(17443\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ P\:0159. 3.3 Rozhodn\:011bte, zda jsou \[CHacek]\[IAcute]sla 224831, \ 20204080301 prvo\[CHacek]\[IAcute]sla\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(PrimeQ[224831]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(PrimeQ[20204080301]\)], "Input"], Cell[BoxData[ \(False\)], "Output"] }, Open ]], Cell["\<\ Opravdu nen\[IAcute]? Pod\[IAcute]vejme se na celo\[CHacek]\[IAcute]seln\ \[EAcute] d\:011blitele tohoto \[CHacek]\[IAcute]sla:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Divisors[20204080301]\)], "Input"], Cell[BoxData[ \({1, 59, 379, 22361, 903541, 53308919, 342442039, 20204080301}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ P\:0159. 3.4 Najd\:011bte nejv\:011bt\[SHacek]\[IAcute] spole\[CHacek]n\ \[YAcute] d\:011blitel, nejmen\[SHacek]\[IAcute] spole\[CHacek]n\[YAcute] n\ \[AAcute]sobek pro \[CHacek]\[IAcute]sla 24268, 31884\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(GCD[24268, 31884]\)], "Input"], Cell[BoxData[ \(4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(LCM[24268, 31884]\)], "Input"], Cell[BoxData[ \(193440228\)], "Output"] }, Open ]], Cell["\<\ P\:0159esv\:011bd\[CHacek]\[IAcute]me se, \:017ee sou\[CHacek]in \ nejv\:011bt\[SHacek]\[IAcute]ho spole\[CHacek]n\[EAcute]ho d\:011blitele a n\ \[AAcute]sobku dvou p\:0159irozen\[YAcute]ch \[CHacek]\[IAcute]sel se rovn\ \[AAcute] sou\[CHacek]inu t\:011bchto \[CHacek]\[IAcute]sel:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(GCD[24268, 31884]*LCM[24268, 31884] == 24268*31884\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ P\:0159. 3.5 Vypi\[SHacek]te v\[SHacek]echny d\:011blitele \ \[CHacek]\[IAcute]sla 2006, 2007\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Divisors[2006]\)], "Input"], Cell[BoxData[ \({1, 2, 17, 34, 59, 118, 1003, 2006}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Divisors[2007]\)], "Input"], Cell[BoxData[ \({1, 3, 9, 223, 669, 2007}\)], "Output"] }, Open ]], Cell["P\:0159\[IAcute]padn\:011b", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[Divisors, {2006, 2007}]\)], "Input"], Cell[BoxData[ \({{1, 2, 17, 34, 59, 118, 1003, 2006}, {1, 3, 9, 223, 669, 2007}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ P\:0159. 3.6 Dokonal\[EAcute] \[CHacek]\[IAcute]slo nazveme takov\[EAcute] p\ \:0159irozen\[EAcute] \[CHacek]\[IAcute]slo, je\:017e je rovno sou\[CHacek]tu \ v\[SHacek]ech sv\[YAcute]ch p\:0159irozen\[YAcute]ch d\:011blitel\:016f men\ \[SHacek]\[IAcute]ch ne\:017e je toto \[CHacek]\[IAcute]slo. Vypi\[SHacek]te v\[SHacek]echna dokonal\[AAcute] \[CHacek]\[IAcute]sla \ men\[SHacek]\[IAcute] ne\:017e 50000\ \>", "Subsubsection"], Cell["\<\ Nap\[IAcute]\[SHacek]eme funkci. kter\[AAcute] se\[CHacek]te v\[SHacek]echny \ d\:011blitele dan\[EAcute]ho \[CHacek]\[IAcute]sla, porovn\[AAcute] s dan\ \[YAcute]m \[CHacek]\[IAcute]slem:\ \>", "Text"], Cell[BoxData[ \(dokonalecislo[\ seznam_] := Select[seznam, 2 # == Plus @@ Divisors[#] &]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(dokonalecislo[Range[49999]]\)], "Input"], Cell[BoxData[ \({6, 28, 496, 8128}\)], "Output"] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["4. 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V tomto p\:0159\[IAcute]pad\:011b je mo\:017en\[EAcute] pou\ \:017e\[IAcute]t PowerExpand. 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\[Rule] \(-1\)}, {x \[Rule] 15}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(P\:0159 . \ 5.22\ \ \ \ \:0158e\[SHacek]te\ v\ R\ \(rovnici : \ \@\(3 x - 7\) \ - \@\(x + 1\)\) = 2\)], "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[\@\(3 x - 7\) - \@\(x + 1\) \[Equal] 2, x]\)], "Input"], Cell[BoxData[ \({{x \[Rule] 4\ \((2 + \@2)\)}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(P\:0159 . \ 5.23\ \ \ \ \:0158e\[SHacek]te\ v\ R\ \(rovnici : \ 2 x\^2 + \@\(2 x\^2 - 4 x + 12\)\) = 4 x + 8\)], "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[2 x\^2 + \@\(2 x\^2 - 4 x + 12\) == 4 x + 8, x]\)], "Input"], Cell[BoxData[ \({{x \[Rule] 1 - \@3}, {x \[Rule] 1 + \@3}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(P\:0159 . \ 5.24\ \ \ \ \:0158e\[SHacek]te\ v\ R\ \(rovnici : \ \ \(\@\(x + 1\) \ - \@\(x - 1\)\)\/\(\@\(x + 1\) + \@\(x - 1\)\)\) = 2 - x\)], "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[\ \(\@\(x + 1\) - \@\(x - 1\)\)\/\(\@\(x + 1\) + \@\(x - 1\)\) == 2 - x, x]\)], "Input"], Cell[BoxData[ \({{x \[Rule] 1}, {x \[Rule] 5\/3}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(P\:0159 . \ 5.25\ \ \ \ \:0158e\[SHacek]te\ v\ R\ \(rovnici : \ 1 + \(a\^2 - 1\)\/x\) = a\)], "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[1 + \(a\^2 - 1\)\/x == a, x]\)], "Input"], Cell[BoxData[ \({{x \[Rule] 1 + a}}\)], "Output"] }, Open ]], Cell["\<\ Syst\[EAcute]m Mathematica nenalezl v\[SHacek]echna \:0159e\[SHacek]en\ \[IAcute], pou\:017eijeme rad\:011bji Reduce:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Reduce[1 + \(a\^2 - 1\)\/x == a, x]\)], "Input"], Cell[BoxData[ \(\((a \[Equal] 1 && x \[NotEqual] 0)\) || \((x \[Equal] 1 + a && 1 + a \[NotEqual] 0)\)\)], "Output"] }, Open ]], Cell["\<\ Zde jsou ji\:017e v\[SHacek]echna \:0159e\[SHacek]en\[IAcute] spr\[AAcute]vn\ \:011b diskutov\[AAcute]na.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ P\:0159. 5.26 \:0158e\[SHacek]te soustavu rovnic: 3x-5y=14 6x-10y=17\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[{3 x - 5 y \[Equal] 14, 6 x - 10 y \[Equal] 17}, {x, y}]\)], "Input"], Cell[BoxData[ \({}\)], "Output"] }, Open ]], Cell["Tedy soustava nem\[AAcute] \:0159e\[SHacek]en\[IAcute].", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ P\:0159. 5.27 \:0158e\[SHacek]te soustavu rovnic: x+4y=37 \[And] \ 2x+5y=53.\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[{x + 4 y \[Equal] 37, 2 x + 5 y \[Equal] 53}, {x, y}]\)], "Input"], Cell[BoxData[ \({{x \[Rule] 9, y \[Rule] 7}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ P\:0159. 5.28 \:0158e\[SHacek]te soustavu rovnic: 2x+3y=12 \[And] \ 4x+6y=24.\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[{2 x + 3 y == 12, 4 x + 6 y == 24}, {x, y}]\)], "Input"], Cell[BoxData[ RowBox[{\(Solve::"svars"\), \(\(:\)\(\ \)\), "\<\"Equations may not give \ solutions for all \\\"solve\\\" variables. \\!\\(\\*ButtonBox[\\\"More\ \[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"Solve::svars\\\"]\\)\"\>"}]], "Message"], Cell[BoxData[ \({{x \[Rule] 6 - \(3\ y\)\/2}}\)], "Output"] }, Open ]], Cell["\<\ Soustava tedy m\[AAcute] nekone\[CHacek]n\:011b mnoho \:0159e\[SHacek]en\ \[IAcute], vztah ve v\[YAcute]sledku ud\[AAcute]v\[AAcute] z\[AAcute]vislost \ x na y.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ P\:0159. 5.29 \:0158e\[SHacek]te soustavu rovnic:x+3\[VerticalSeparator]y\ \[VerticalSeparator]\[Equal]1 \[And] x+y+3\[Equal]0.\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[{x + 3 Abs[y] \[Equal] 1, x + y + 3 \[Equal] 0}, {x, y}]\)], "Input"], Cell[BoxData[ RowBox[{\(Solve::"ifun"\), \(\(:\)\(\ \)\), "\<\"Inverse functions are \ being used by \\!\\(Solve\\), so some solutions may not be found; use Reduce \ for complete solution information. \ 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Soustavu vy\:0159e\[SHacek]\[IAcute]me po\[CHacek]etn\:011b pomoc\[IAcute] \ Solve. Nezn\[AAcute]m\[EAcute] x, y nemus\[IAcute]me ani uv\[AAcute]d\:011bt: \ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[{\ 4 x\^2 + y\^2 == 4, x\^2 + 4 y\^2 == 4}]\)], "Input"], Cell[BoxData[ \({{x \[Rule] \(-\(2\/\@5\)\), y \[Rule] \(-\(2\/\@5\)\)}, {x \[Rule] \(-\(2\/\@5\)\), y \[Rule] 2\/\@5}, {x \[Rule] 2\/\@5, y \[Rule] \(-\(2\/\@5\)\)}, {x \[Rule] 2\/\@5, y \[Rule] 2\/\@5}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(P\:0159 . \ 5.34\ \ \ \:0158e\[SHacek]te\ soustavu\ \(rovnic : x\^2\/a\^2 + y\^2\/b\^2\) = \(1\ \[And] \ y = k\ x\)\)], "Subsubsection"], Cell["\<\ Zde mus\[IAcute]me v Solve uv\[EAcute]st vzhledem ke kter\[YAcute]m nezn\ \[AAcute]m\[YAcute]m soustavu rovnic \:0159e\[SHacek]\[IAcute]me.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[{x\^2\/a\^2 + y\^2\/b\^2 == 1, \ y == k*x}, {x, y}]\)], "Input"], Cell[BoxData[ \({{y \[Rule] \(-\(\(a\ b\ k\)\/\@\(b\^2 + a\^2\ k\^2\)\)\), x \[Rule] \(-\(\(a\ b\)\/\@\(b\^2 + a\^2\ k\^2\)\)\)}, {y \[Rule] \(a\ \ b\ k\)\/\@\(b\^2 + a\^2\ k\^2\), x \[Rule] \(a\ b\)\/\@\(b\^2 + a\^2\ k\^2\)}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(P\:0159 . \ 5.35\ \ \ V\ mno\:017ein\:011b\ R\ \:0159e\[SHacek]te\ \(nerovnici : \ \ \((x - 3)\)\^2 + \((x + 1)\)\^2 \[LessEqual] 2 x\^2 - 6 x + 13\)\)], "Subsubsection"], Cell[BoxData[ \(<< Algebra`InequalitySolve`\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(InequalitySolve[\((x - 3)\)\^2 + \((x + 1)\)\^2 \[LessEqual] 2 x\^2 - 6 x + 13, x]\)], "Input"], Cell[BoxData[ \(x \[LessEqual] 3\/2\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(P\:0159 . \ 5.36\), " ", "V", " ", "mno\:017ein\:011b", " ", "R", " ", "\:0159e\[SHacek]te", " ", RowBox[{"nerovnici", ":", " ", RowBox[{\(x\/\[LeftBracketingBar]x - 1\[RightBracketingBar] \ \[GreaterEqual] 0\), StyleBox["\[And]", FontSize->9], \(1\/x < x + 1\)}]}]}]], "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(InequalitySolve[x/Abs[x - 1] \[GreaterEqual] 0 && 1/x < x + 1, x]\)], "Input"], Cell[BoxData[ \(1\/2\ \((\(-1\) + \@5)\) < x < 1 || x > 1\)], "Output"] }, Open ]], Cell["\<\ Nerovnice \:0159e\[SHacek]en\[AAcute] pomoc\[IAcute] InequalitySolve tedy m\ \:016f\:017ee obsahovat i absolutn\[IAcute] hodnoty\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(P\:0159 . \ 5.37\ \ V\ mno\:017ein\:011b\ R\ \:0159e\[SHacek]te\ \(nerovnici : 1\/x + 1\/x\^2 \[GreaterEqual] 1\)\)], "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(InequalitySolve[1\/x + 1\/x\^2 \[GreaterEqual] 1, x]\)], "Input"], Cell[BoxData[ \(1\/2\ \((1 - \@5)\) \[LessEqual] x < 0 || 0 < x \[LessEqual] 1\/2\ \((1 + \@5)\)\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Numerick\[EAcute] \:0159e\[SHacek]en\[IAcute] rovnic", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(P\:0159 . \ 5.38\ \ \:0158e\[SHacek]te\ rovnici\ x\^5 + x\^4 - 4 x\^3 + 2 x\^2 - 3 x - 7 = 0\)], "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Solve[x\^5 + x\^4 - 4 x\^3 + 2 x\^2 - 3 x - 7 \[Equal] 0, x]\)\(\ \)\)\)], "Input"], Cell[BoxData[ \({{x \[Rule] Root[\(-7\) - 3\ #1 + 2\ #1\^2 - 4\ #1\^3 + #1\^4 + #1\^5 &, 1]}, {x \[Rule] Root[\(-7\) - 3\ #1 + 2\ #1\^2 - 4\ #1\^3 + #1\^4 + #1\^5 &, 2]}, {x \[Rule] Root[\(-7\) - 3\ #1 + 2\ #1\^2 - 4\ #1\^3 + #1\^4 + #1\^5 &, 3]}, {x \[Rule] Root[\(-7\) - 3\ #1 + 2\ #1\^2 - 4\ #1\^3 + #1\^4 + #1\^5 &, 4]}, {x \[Rule] Root[\(-7\) - 3\ #1 + 2\ #1\^2 - 4\ #1\^3 + #1\^4 + #1\^5 &, 5]}}\)], "Output"] }, Open ]], Cell["\<\ To n\[AAcute]m mnoho nepomohlo, algebraick\[EAcute] rovnice stupn\:011b vy\ \[SHacek]\[SHacek]\[IAcute]ho ne\:017e 4 Mathematica (a jak v\[IAcute]me \ nejen ona) \:0159e\[SHacek]it obecn\:011b neum\[IAcute]. Pou\:017eijeme tedy \ jin\[EAcute] metody - numerick\[EAcute]. Pou\:017eijeme postupn\:011b NRoots, \ NSolve, po sestrojen\[IAcute] grafu FindRoot.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NRoots[x\^5 + x\^4 - 4 x\^3 + 2 x\^2 - 3 x - 7 == 0, x]\)], "Input"], Cell[BoxData[ \(x \[Equal] \(-2.744632420104718`\) || x \[Equal] \(-0.8808584760078594`\) || x \[Equal] \(\(0.4145201849499447`\)\(\[InvisibleSpace]\)\) - 1.1999598403356018`\ \[ImaginaryI] || x \[Equal] \(\(0.4145201849499447`\)\(\[InvisibleSpace]\)\) + 1.1999598403356018`\ \[ImaginaryI] || x \[Equal] 1.7964505262126877`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?||\)\)], "Input"], Cell[BoxData[ RowBox[{"\<\"e1 || e2 || ... is the logical OR function. 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