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| Original file line number | Diff line number | Diff line change |
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| @@ -0,0 +1,219 @@ | ||
| --- | ||
| created: 2024-11-12T14:31:11 | ||
| up: "[[📖NI-MPI]]" | ||
| --- | ||
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| TARGET DECK: NI-MPI | ||
| FILE TAGS: NI-MPI cviceni08 status-toReview | ||
|
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| ### Strojová čísla | ||
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| ### Cvičení 15.1 | ||
|  | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & \\ | ||
| & \text{a) -1/13} \\ | ||
| & a = (-1)^{s} \cdot \textcolor{Skyblue}{1. m} \cdot \textcolor{Red}{2^{e-127}} \\ | ||
| & \\ | ||
| & \text{Mocniny 2} \\ | ||
| & 2^{3} = 8 \\ | ||
| & 2^{2} = 4 \\ | ||
| & 2^{1} = 2 \\ | ||
| & 2^{0} = 1 \\ | ||
| & 2^{-1} = \tfrac{1}{2} \\ | ||
| & 2^{-2} = \tfrac{1}{4} \\ | ||
| & 2^{-3} = \tfrac{1}{8} \\ | ||
| & \dots \\ | ||
| & \\ | ||
| & \text{Znaménko} \\ | ||
| & s = 1 \quad \text{\small{Pro záporné číslo}} \\ | ||
| & \\ | ||
| & \text{Počítání mantisy} \\ | ||
| & a_{-1} =\tfrac{1}{13} : \tfrac{1}{2} = \lfloor \tfrac{2}{13} \rfloor = 0 \\ | ||
| & a_{-2} =\tfrac{2}{13} :\tfrac{1}{2} =\lfloor \tfrac{4}{13} \rfloor =0 \\ | ||
| & a_{-3} =\lfloor \tfrac{8}{13} \rfloor = 0 \\ | ||
| & a_{-4} = \lfloor \tfrac{16}{13} \rfloor = 1 \\ | ||
| & a_{-5} = \lfloor \tfrac{6}{13} \rfloor = 0 \\ | ||
| & a_{-6} = \lfloor \tfrac{12}{13} \rfloor = 0 \\ | ||
| & a_{-7} = \lfloor \tfrac{24}{13} \rfloor = 1 \\ | ||
| & a_{-8} = \lfloor \tfrac{22}{13} \rfloor = 1 \\ | ||
| & a_{-9} = \lfloor \tfrac{18}{13} \rfloor = 1 \\ | ||
| & a_{-10} = \lfloor \tfrac{10}{13} \rfloor = 0 \\ | ||
| & a_{-11} = \lfloor \tfrac{20}{13} \rfloor = 1 \\ | ||
| & a_{-12} = \lfloor \tfrac{14}{13} \rfloor =1 \\ | ||
| & a_{-13} = a_{-1} \quad \text{\small{Zde jsme se zacyklili}} \\ | ||
| & \tfrac{1}{13} = 0.\overline{000100111011} \\ | ||
| & = 1.\overline{001110110001} \cdot 2^{-4} \quad \text{\small{Znormovali jsme to}} \\ | ||
| & \\ | ||
| & \text{\small{Mantisa tedy bude:}} \\ | ||
| & m = 00111011000100111011000 \quad \text{\small{23 číslic}} \\ | ||
| & \\ | ||
| & \text{Spočteme exponent} \\ | ||
| & -4 = e - 127 \\ | ||
| & e = 123 \\ | ||
| & 123 : 128 = 0 \quad 123 \\ | ||
| & 123 : 64 = 1 \quad 59 \\ | ||
| & 59 : 32 = 1 \quad 27 \\ | ||
| & 27 : 16 = 1 \quad 11 \\ | ||
| & 11 : 8 = 1 \quad 3 \\ | ||
| & 3 : 4 = 0 \quad 3 \\ | ||
| & 3: 2 = 1 \quad 1 \\ | ||
| & 1 : 1 = 0 \quad 0 \\ | ||
| & e = 01111011 | ||
| \end{align*} | ||
| $$ | ||
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| ### Cvičení 15.1 b) | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & \tfrac{1}{17} \\ | ||
| & \\ | ||
| & \text{Znaménko} \\ | ||
| & s = 0 \\ | ||
| & \\ | ||
| & \text{Mantisa} \\ | ||
| & a_{-1} = \tfrac{1}{17} : \tfrac{1}{2} = \lfloor \tfrac{2}{17} \rfloor = 0 \\ \\ | ||
| & \text{\small{Ty jednotlivé kroky se dají počítat z hlavy:}} \\ | ||
| & \tfrac{1}{17} = 0.\overline{00001111} \\ | ||
| & = 1.\overline{11100001} \cdot 2^{-5} \quad \text{\small{normalizace}} \\ | ||
| & \\ | ||
| & \text{Exponent} \\ | ||
| & -5 = e-127 \\ | ||
| & e=122 = \left(01111010 \right)_{2} \\ | ||
| & \\ | ||
| & \text{Celkem tedy} \\ | ||
| & s = 0 \\ | ||
| & e = 01111010 \\ | ||
| & m = 11100001111000011110000 \\ | ||
| \end{align*} | ||
| $$ | ||
|
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| ### Cvičení 15.1 c) | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & -\tfrac{1}{13} +\tfrac{1}{17} = - \left(\tfrac{1}{13} - \tfrac{1}{17} \right) \\ | ||
| & \textcolor{Lime}s = 1 \\ | ||
| & \\ | ||
| & \text{Sčítání} \\ | ||
| & \tfrac{1}{13} = 1.00111011000100111011000 \cdot 2^{-4} \\ | ||
| & -\tfrac{1}{17} = -0.111100001111000011110000 \cdot 2^{-4} \quad \text{\small{Převedli jsme to na stejný exponent šoupnutím doprava}} \\ | ||
| & \quad \ \ 1.00111011000100111011000 \cdot 2^{-4} \\ | ||
| & \underline{-0.111100001111000011110000} \cdot 2^{-4} \quad \text{\small{Odčítání pod sebou}} \\ | ||
| & \quad 0.01001010001000101100000 \cdot 2^{-4} \\ | ||
| & \\ | ||
| & \textcolor{Blue}m = 001010001000101100000\textcolor{Orange}{00} \quad \text{\small{Poslední 2 bity jsme doplnili pomocí 0}} \\ | ||
| & \\ | ||
| & \text{Exponent} \\ | ||
| & \textcolor{Orange}e - 127 = -6 \\ | ||
| & \textcolor{Orange}e = 121 \\ | ||
| & \textcolor{Orange}e = 01111001_{2} \\ | ||
| & \\ | ||
| & \text{Celkem tedy} \\ | ||
| & \underset{\textcolor{Lime}s}{\textcolor{Lime}{0}}\underset{\textcolor{Orange}e}{\textcolor{Orange}{01111001}}\underset{\textcolor{Blue}m}{\textcolor{Blue}{001010001000101100000}} \\ | ||
| & | ||
| \end{align*} | ||
| $$ | ||
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| ### Grupoid, monoid, pologrupa, grupa | ||
|  | ||
| ### Cvičení 16.2a | ||
|  | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & \text{Prosté funkce} \\ | ||
| & M = \set{f(x), f \text{\small{je prostá}}, D_{f}= \mathbb{R}} \\ | ||
| & a) f \circ g = f(g(x)) \\ | ||
| & \\ | ||
| & \text{Uzavřenost} f,g \in M \text{\small{?}} \\ | ||
| & \text{\small{Prostá funkce:}} \\ | ||
| & x_{1} \neq x_{2} \Rightarrow f(x_{1}) \neq f(x_{2}) \\ | ||
| & x_{1} \neq x_{2} \underbrace{\Rightarrow}_{g \text{\small{ prostá}} } g(x_{1}) \neq g(x_{2}) \underbrace{\Rightarrow}_{f \text{\small{ prostá}}} f(g(x_{1})) \neq f(g(x_{2})) \\ | ||
| & (f \circ g)(x_{1}) \neq (f \circ g)(x_{2}) \\ | ||
| & (f \circ g)\text{\small{je prostá funkce}} \\ | ||
| & D_{(f \circ g)} = \mathbb{R} \Rightarrow f \circ g \in M \\ | ||
| \end{align*} | ||
| $$ | ||
|
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| ### Cvičení 16.2b | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & f^{-'} \in M \\ | ||
| & \text{\small{protipříklad}} \ln{x} \\ | ||
| & D_{\ln{x}} \neq \mathbb{R} \Rightarrow M \text{\small{není uzavřené vůči}} f^{-1} \\ | ||
| \end{align*} | ||
| $$ | ||
|
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| ### Cvičení 16.2c | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & \underbrace{e(x) = x}_{\text{\small{Neutrální prvek}} } \quad e \circ f = f \circ e = f \\ | ||
| & f \circ (g \circ h) = (f \circ g) \circ h \\ | ||
| & f \circ g(h(x)) = f(g(h(x))) = (f \circ g)(h(x)) = (f \circ g) \circ h \\ | ||
| & \Rightarrow \text{\small{Asociativnost platí}} \\ | ||
| & \\ | ||
| & \text{Závěr} \\ | ||
| & (M, \circ) \text{\small{ je monoid.}} | ||
| \end{align*} | ||
| $$ | ||
|
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| ### Cvičení 16.6 | ||
|  | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & P = \set{p(x) = a_{0} + a_{1}x + a_{2} x^2 +a_{3}x^3 +\dots} \\ | ||
| & p'(x) = a_{1} + 2a_{2} x + 3a_{3}x^2 +\dots \in P \\ | ||
| & \Rightarrow \text{\small{Ano, P je uzavřená vzhledem k derivaci}} | ||
| \end{align*} | ||
| $$ | ||
|
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| ### Cvičení 16.7 | ||
|  | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & \left(\set{1} , : \right) \text{\small{ je grupa, ale my chceme větší}} \\ | ||
| & \\ | ||
| & \text{Jak bude vypadat ta množina} \\ | ||
| & \left(\set{1, -1} , : \right) \\ | ||
| & \text{\small{Zkusíme uzavřenost:}} \\ | ||
| & 1:1=1 \\ | ||
| & 1:(-1)=-1 \\ | ||
| & -1:1=-1 \\ | ||
| & -1:-1=1 \\ | ||
| & \\ | ||
| & \text{\small{Zkusíme asociativnost}} \\ | ||
| & a \circ (b \circ c) = (a \circ b) \circ c \\ | ||
| & \text{\small{Bude platit, jde jen o počet kladných a záporných jedniček}} \\ | ||
| & \\ | ||
| & \text{\small{Zkusíme neutrální prvek}} \\ | ||
| & e=1 \quad e \circ a = a \circ e = a \\ | ||
| & \\ | ||
| & \text{\small{Je to grupa?}} \\ | ||
| & (1)^{-1}=1 \\ | ||
| & (-1)^{-1} = -1 \\ | ||
| & \\ | ||
| & \text{\small{Je to Abelovská grupa?}} \\ | ||
| & a \circ b = b \circ a \\ | ||
| & 1:-1=-1:1=-1 \\ | ||
| & | ||
| \end{align*} | ||
| $$ | ||
| ### Cvičení 18.8 | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & M \subset \mathbb{R} \\ | ||
| & M = \set{1,2,3, \dots ,0,-1,-2,-3, \dots } = \mathbb{Z} \\ | ||
| & (\mathbb{Z},+) \text{\small{je grupa}} | ||
| \end{align*} | ||
| $$ |