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authorAnton Luka Šijanec <anton@sijanec.eu>2024-01-05 19:46:28 +0100
committerAnton Luka Šijanec <anton@sijanec.eu>2024-01-05 19:46:28 +0100
commitd2e03945e00795d68fca77e1c1978376d06a3156 (patch)
treef8892b6a986c1c2db7e893618094e7491d0b5f09
parentMerge branch 'master' of ssh://ni/var/lib/git/sijanec/r (diff)
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Diffstat (limited to '')
-rwxr-xr-xšola/aps1/funkcijebin0 -> 20528 bytes
-rw-r--r--šola/la/dn4/dokument.lyx2272
-rw-r--r--šola/la/dn4/gcd.r29
-rw-r--r--šola/p1/dn/DN06_izhodisce/Premica.java68
-rw-r--r--šola/p1/dn/DN06_izhodisce/Test01.java85
-rw-r--r--šola/p1/dn/DN06_izhodisce/Tocka.java40
6 files changed, 2494 insertions, 0 deletions
diff --git a/šola/aps1/funkcije b/šola/aps1/funkcije
new file mode 100755
index 0000000..0c36f6b
--- /dev/null
+++ b/šola/aps1/funkcije
Binary files differ
diff --git a/šola/la/dn4/dokument.lyx b/šola/la/dn4/dokument.lyx
new file mode 100644
index 0000000..2942005
--- /dev/null
+++ b/šola/la/dn4/dokument.lyx
@@ -0,0 +1,2272 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass article
+\begin_preamble
+\usepackage{siunitx}
+\usepackage{pgfplots}
+\usepackage{listings}
+\usepackage{multicol}
+\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}}
+\usepackage{amsmath}
+\usepackage{tikz}
+\newcommand{\udensdash}[1]{%
+ \tikz[baseline=(todotted.base)]{
+ \node[inner sep=1pt,outer sep=0pt] (todotted) {#1};
+ \draw[densely dashed] (todotted.south west) -- (todotted.south east);
+ }%
+}%
+\end_preamble
+\use_default_options true
+\begin_modules
+enumitem
+theorems-ams
+\end_modules
+\maintain_unincluded_children false
+\language slovene
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
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+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification false
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1cm
+\topmargin 0cm
+\rightmargin 1cm
+\bottommargin 2cm
+\headheight 1cm
+\headsep 1cm
+\footskip 1cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style german
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Title
+Rešitev četrte domače naloge Linearne Algebre
+\end_layout
+
+\begin_layout Author
+
+\noun on
+Anton Luka Šijanec
+\end_layout
+
+\begin_layout Date
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+today
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Abstract
+Za boljšo preglednost sem svoje rešitve domače naloge prepisal na računalnik.
+ Dokumentu sledi še rokopis.
+ Naloge je izdelala asistentka Ajda Lemut.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+newcommand
+\backslash
+euler{e}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Reši enačbo
+\begin_inset Formula
+\[
+\left|\begin{array}{cccc}
+1 & 2 & 3 & 4\\
+x+1 & 2 & x+3 & 4\\
+1 & x+2 & x+4 & x+5\\
+1 & -3 & -4 & -5
+\end{array}\right|=\left|\begin{array}{cc}
+3x & -1\\
+6 & x+1
+\end{array}\right|
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Paragraph
+Rešitev
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula
+\[
+\left|\begin{array}{cc}
+3x & -1\\
+6 & x+1
+\end{array}\right|=3x\left(x+1\right)+6=3x^{2}+3x+6
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula
+\[
+\left|\begin{array}{cccc}
+1 & 2 & 3 & 4\\
+x+1 & 2 & x+3 & 4\\
+1 & x+2 & x+4 & x+5\\
+1 & -3 & -4 & -5
+\end{array}\right|=\left|\begin{array}{cccc}
+1 & 2 & 3 & 4\\
+x+1 & 2 & x+3 & 4\\
+1 & x+2 & x+4 & x+5\\
+0 & -5 & -7 & -9
+\end{array}\right|=
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula
+\[
+=\left|\begin{array}{cccc}
+0 & -x & -x-1 & -x-1\\
+x+1 & 2 & x+3 & 4\\
+1 & x+2 & x+4 & x+5\\
+0 & -5 & -7 & -9
+\end{array}\right|=
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula
+\[
+=-\left(x+1\right)\left|\begin{array}{ccc}
+-x & -x-1 & -x-1\\
+x+2 & x+4 & x+5\\
+-5 & -7 & -9
+\end{array}\right|+\left|\begin{array}{ccc}
+-x & -x-1 & -x-1\\
+2 & x+3 & 4\\
+-5 & -7 & -9
+\end{array}\right|=
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula
+\[
+=x-1+4x^{2}-5x+1=4x^{2}-6x
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+4x^2-6x&=3x^2+3x+6
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+x^2-9x-6&=0
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula
+\[
+x_{1,2}=\frac{9\pm\sqrt{81+24}}{2}=\frac{9\pm\sqrt{105}}{2},\quad x_{1}=\frac{9+\sqrt{105}}{2},x_{2}=\frac{9-\sqrt{105}}{2}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Dokaži, da je preslikava
+\begin_inset Formula $x\mapsto x^{-1}$
+\end_inset
+
+ avtomorfizem grupe natanko tedaj, ko je grupa komutativna.
+\end_layout
+
+\begin_deeper
+\begin_layout Paragraph
+Rešitev
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+udensdash{$f
+\backslash
+left(x
+\backslash
+right)=x^{-1}
+\backslash
+text{ je avtomorfizem}
+\backslash
+Longleftrightarrow
+\backslash
+forall a,b
+\backslash
+in M:a
+\backslash
+cdot b=b
+\backslash
+cdot a$}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Dokaz
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset CommandInset label
+LatexCommand label
+name "enu:Enota-se-preslika"
+
+\end_inset
+
+Enota se preslika v enoto.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+e
+\backslash
+cdot e^{-1}&=e&&
+\backslash
+text{(definicija inverza $a
+\backslash
+cdot a^{-1}=e$)}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+e
+\backslash
+cdot e^{-1}&=e^{-1}&&
+\backslash
+text{(definicija enote $e
+\backslash
+cdot a=a$)}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+$$
+\backslash
+Longrightarrow e=e
+\backslash
+cdot e^{-1}=e^{-1}$$
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset CommandInset label
+LatexCommand label
+name "enu:Da-je-preslikava"
+
+\end_inset
+
+Da je preslikava bijektivna, moramo dokazati, da je injektivna, torej, da
+ so v komutativni grupi inverzi enolični — da dva različna elementa nimata
+ istega inverza, in da je surjektivna, torej, da je kodomena enaka zalogi
+ vrednosti.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Naj bo
+\begin_inset Formula $\left(M,\cdot\right)$
+\end_inset
+
+ grupa.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+udensdash{$
+\backslash
+forall a,b
+\backslash
+in M:
+\backslash
+left(a^{-1}=b^{-1}
+\backslash
+Longrightarrow a=b
+\backslash
+right)$}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Naj bo
+\begin_inset Formula $a^{-1}=b^{-1}$
+\end_inset
+
+.
+
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+udensdash{$a=b$}
+\end_layout
+
+\end_inset
+
+
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+a
+\backslash
+cdot a^{-1}&=e&&b
+\backslash
+cdot b^{-1}=e
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+a
+\backslash
+cdot b^{-1}&=e&&/
+\backslash
+cdot b
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+a
+\backslash
+cdot e&=e
+\backslash
+cdot b
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+$$a=b$$
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset CommandInset label
+LatexCommand label
+name "enu:Dokaz-ohranjanja-inverzov:"
+
+\end_inset
+
+Dokaz ohranjanja inverzov:
+\begin_inset Formula $f\left(x\right)^{-1}=f\left(x^{-1}\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula
+\[
+\left(x^{-1}\right)^{-1}=\left(x^{-1}\right)^{-1}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Ob upoštevanju
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "enu:Da-je-preslikava"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+ je to enako kot
+\begin_inset Formula $x=x$
+\end_inset
+
+, kar drži, torej je preslikava injektivna.
+\end_layout
+
+\begin_layout Standard
+Da je surjektivna, mora veljati
+\begin_inset Formula $\forall x^{-1}\exists x:x^{-1}=x$
+\end_inset
+
+.
+ Naj bo tak
+\begin_inset Formula $x$
+\end_inset
+
+ kar
+\begin_inset Formula $\left(x^{-1}\right)^{-1}$
+\end_inset
+
+.
+ Dokažimo:
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+udensdash{$
+\backslash
+left(x^{-1}
+\backslash
+right)^{-1}=x$}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+left(x^{-1}
+\backslash
+right)^{-1}&
+\backslash
+overset{?}{=}x&&/
+\backslash
+cdot x^{-1}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+e&=e
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+Torej je preslikava bijektivna.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset CommandInset label
+LatexCommand label
+name "enu:Asociativnost-operacije."
+
+\end_inset
+
+Asociativnost operacije.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Zahtevamo, da operacija ostane enaka, zato je asociativna.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset CommandInset label
+LatexCommand label
+name "enu:Po-definiciji-homomorfizma"
+
+\end_inset
+
+Po definiciji homomorfizma je treba dokazati, da
+\begin_inset Formula
+\[
+\forall a,b\in M:\left(f\left(a\cdot_{1}b\right)=f\left(a\right)\cdot_{2}f\left(b\right)\right)\Longleftrightarrow\text{grupa je Abelova}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Naj bosta
+\begin_inset Formula $a,b$
+\end_inset
+
+ poljubna iz grupe
+\begin_inset Formula $\left(M,\cdot\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Lemma
+V grupi
+\begin_inset Formula $\left(N,\circ\right)$
+\end_inset
+
+ velja za poljubna
+\begin_inset Formula $x,y\in N$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Lemma
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+udensdash{$
+\backslash
+left(a
+\backslash
+circ b
+\backslash
+right)^{-1}=y^{-1}
+\backslash
+circ x^{-1}$}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Lemma
+Dokaz leme:
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+left(x
+\backslash
+circ y
+\backslash
+right)
+\backslash
+circ
+\backslash
+backslash&&
+\backslash
+left(x
+\backslash
+circ y
+\backslash
+right)^{-1}&
+\backslash
+overset{?}{=}y^{-1}
+\backslash
+circ x^{-1}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&&
+\backslash
+left(x
+\backslash
+circ y
+\backslash
+right)
+\backslash
+circ
+\backslash
+left(x
+\backslash
+circ y
+\backslash
+right)^{-1}&
+\backslash
+overset{?}{=}
+\backslash
+left(x
+\backslash
+circ y
+\backslash
+right)
+\backslash
+circ
+\backslash
+left(y^{-1}
+\backslash
+circ x^{-1}
+\backslash
+right)=x
+\backslash
+circ
+\backslash
+left(y
+\backslash
+circ y^{-1}
+\backslash
+right)
+\backslash
+circ x^{-1}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&&e&
+\backslash
+overset{?}{=}x
+\backslash
+circ e
+\backslash
+circ x^{-1}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&&e&
+\backslash
+overset{?}{=}x
+\backslash
+circ x^{-1}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&&e&=e
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\end_inset
+
+Konec leme — lema je dokazana.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+f
+\backslash
+left(a
+\backslash
+cdot b
+\backslash
+right)&
+\backslash
+overset{?}{=}f
+\backslash
+left(a
+\backslash
+right)
+\backslash
+cdot f
+\backslash
+left(b
+\backslash
+right)
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+b^{-1}
+\backslash
+cdot a^{-1}
+\backslash
+overset{
+\backslash
+text{lema}}{=}
+\backslash
+left(a
+\backslash
+cdot b
+\backslash
+right)^{-1}&
+\backslash
+overset{?}{=}a^{-1}
+\backslash
+cdot b^{-1}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+b^{-1}
+\backslash
+cdot a^{-1}&=a^{-1}
+\backslash
+cdot b^{-1}&&
+\backslash
+text{velja natanko tedaj, ko je grupa Abelova.}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "enu:Enota-se-preslika"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+,
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "enu:Da-je-preslikava"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+,
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "enu:Dokaz-ohranjanja-inverzov:"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+,
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "enu:Asociativnost-operacije."
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+ veljajo ne glede na to, ali je grupa komutativna ali ne,
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "enu:Po-definiciji-homomorfizma"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+ pa velja natanko tedaj, ko je grupa komutativna.
+\begin_inset Formula $\qed$
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Prepričaj se, da je množica
+\begin_inset Formula $\mathbb{Z}\times\mathbb{Z}$
+\end_inset
+
+ komutativen kolobar za operaciji
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+left(a, b
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(c, d
+\backslash
+right)&=
+\backslash
+left(a+c, b+d
+\backslash
+right)
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+left(a, b
+\backslash
+right)
+\backslash
+otimes
+\backslash
+left(c, d
+\backslash
+right)&=
+\backslash
+left(ac, bd
+\backslash
+right)
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\end_inset
+
+Poišči tudi vse delitelje niča, tj.
+ neničelne elemente
+\begin_inset Formula $\left(a,b\right)$
+\end_inset
+
+, da velja
+\begin_inset Formula $\left(a,b\right)\otimes\left(c,d\right)=0\left(=e_{\oplus}\right)$
+\end_inset
+
+ za nek neničeln
+\begin_inset Formula $\left(c,d\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Paragraph
+Rešitev
+\end_layout
+
+\begin_layout Itemize
+Dokažimo distributivnost!
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+otimes
+\backslash
+left(
+\backslash
+left(c,d
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(e,f
+\backslash
+right)
+\backslash
+right)&
+\backslash
+overset{?}{=}
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+otimes
+\backslash
+left(c,d
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+otimes
+\backslash
+left(e,f
+\backslash
+right)
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+otimes
+\backslash
+left(c+e,d+f
+\backslash
+right)&
+\backslash
+overset{?}{=}
+\backslash
+left(ac,bd
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(ae,bf
+\backslash
+right)
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+left(a
+\backslash
+cdot
+\backslash
+left(c+e
+\backslash
+right),b
+\backslash
+cdot
+\backslash
+left(d+f
+\backslash
+right)
+\backslash
+right)&=
+\backslash
+left(a
+\backslash
+cdot c+a
+\backslash
+cdot e,b
+\backslash
+cdot d+b
+\backslash
+cdot f
+\backslash
+right)
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+Velja, ker je $
+\backslash
+left(
+\backslash
+mathbb{Z},+,
+\backslash
+cdot
+\backslash
+right)$ distributiven bigrupoid.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Dokažimo
+\begin_inset Formula $\left(\mathbb{Z}\times\mathbb{Z},\oplus\right)$
+\end_inset
+
+ je Abelova grupa!
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+Komutativnost:
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+forall
+\backslash
+left(a,b
+\backslash
+right),
+\backslash
+left(c,d
+\backslash
+right)
+\backslash
+in
+\backslash
+mathbb{Z}
+\backslash
+times
+\backslash
+mathbb{Z}:&&
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(c,d
+\backslash
+right)&=
+\backslash
+left(c,d
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&&
+\backslash
+left(a+c,b+d
+\backslash
+right)&=
+\backslash
+left(c+a,d+b
+\backslash
+right)
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+Velja, ker je $
+\backslash
+left(
+\backslash
+mathbb{Z},+
+\backslash
+right)$ komutativen grupoid.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Notranja operacija:
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+forall
+\backslash
+left(a,b
+\backslash
+right),
+\backslash
+left(c,d
+\backslash
+right)
+\backslash
+in
+\backslash
+mathbb{Z}
+\backslash
+times
+\backslash
+mathbb{Z}:&&
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(c,d
+\backslash
+right)&
+\backslash
+in
+\backslash
+mathbb{Z}
+\backslash
+times
+\backslash
+mathbb{Z}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&&
+\backslash
+left(a+c,b+d
+\backslash
+right)&
+\backslash
+in
+\backslash
+mathbb{Z}
+\backslash
+times
+\backslash
+mathbb{Z}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+Velja, ker je $
+\backslash
+left(
+\backslash
+mathbb{Z},+
+\backslash
+right)$ grupoid.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Asociativnost:
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+forall
+\backslash
+left(a,b
+\backslash
+right),
+\backslash
+left(c,d
+\backslash
+right),
+\backslash
+left(e,f
+\backslash
+right)
+\backslash
+in
+\backslash
+mathbb{Z}
+\backslash
+times
+\backslash
+mathbb{Z}:&&
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(
+\backslash
+left(c,d
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(e,f
+\backslash
+right)
+\backslash
+right)&=
+\backslash
+left(
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(c,d
+\backslash
+right)
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(e,f
+\backslash
+right)
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&&
+\backslash
+left(a+
+\backslash
+left(c+e
+\backslash
+right),b+
+\backslash
+left(d,f
+\backslash
+right)
+\backslash
+right)&=
+\backslash
+left(
+\backslash
+left(a+c
+\backslash
+right)+e,
+\backslash
+left(b+d
+\backslash
+right)+f
+\backslash
+right)
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+Velja, ker je $
+\backslash
+left(
+\backslash
+mathbb{Z},+
+\backslash
+right)$ grupoid.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Enota:
+\begin_inset Formula
+\[
+\exists e\in\mathbb{Z}\times\mathbb{Z}\ni:\forall\left(a,b\right)\in\mathbb{Z}\times\mathbb{Z}:\left(a,b\right)\oplus e=\left(a,b\right)
+\]
+
+\end_inset
+
+naj bo
+\begin_inset Formula $e\coloneqq\left(0,0\right)$
+\end_inset
+
+
+\begin_inset Formula
+\[
+\left(a,b\right)\oplus\left(0,0\right)=\left(a+b,b+0\right)=\left(a,b\right)
+\]
+
+\end_inset
+
+Velja, ker je
+\begin_inset Formula $0$
+\end_inset
+
+ enota v
+\begin_inset Formula $\left(\mathbb{Z},+\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Inverzi:
+\begin_inset Formula
+\[
+\forall\left(a,b\right)\in\mathbb{Z}\times\mathbb{Z}\exists t\in\mathbb{Z}\times\mathbb{Z}\ni:\left(a,b\right)\oplus t=e_{\oplus}=\left(0,0\right)
+\]
+
+\end_inset
+
+naj bo
+\begin_inset Formula $t\coloneqq\left(-a,-b\right)$
+\end_inset
+
+
+\begin_inset Formula
+\[
+\left(a,b\right)\oplus\left(-a,-b\right)=\left(a-a,b-b\right)=\left(0,0\right)=e_{\oplus}
+\]
+
+\end_inset
+
+Velja, ker je
+\begin_inset Formula $\left(\mathbb{Z},+\right)$
+\end_inset
+
+ grupa.
+\end_layout
+
+\end_deeper
+\begin_layout Itemize
+Dokažimo komutativnost
+\begin_inset Formula $\left(\mathbb{Z}\times\mathbb{Z},\otimes\right)$
+\end_inset
+
+!
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+forall
+\backslash
+left(a,b
+\backslash
+right),
+\backslash
+left(c,d
+\backslash
+right)
+\backslash
+in
+\backslash
+mathbb{Z}
+\backslash
+times
+\backslash
+mathbb{Z}:&&
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+otimes
+\backslash
+left(c,d
+\backslash
+right)&
+\backslash
+overset{?}{=}
+\backslash
+left(c,d
+\backslash
+right)
+\backslash
+otimes
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&&
+\backslash
+left(ac,bd
+\backslash
+right)=
+\backslash
+left(ca,db
+\backslash
+right)
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+Velja, ker je $
+\backslash
+left(
+\backslash
+mathbb{Z},
+\backslash
+cdot
+\backslash
+right)$ komutativen grupoid.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula
+\[
+\qed
+\]
+
+\end_inset
+
+Vsi delitelji niča
+\begin_inset Formula $=\left\{ \left(a,b\right)\in\mathbb{Z}\times\mathbb{Z};\left(a,b\right)\otimes\left(c,d\right)=e_{\oplus}=\left(0,0\right)\right\} $
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Itemize
+Če je
+\begin_inset Formula $c=0$
+\end_inset
+
+ in
+\begin_inset Formula $d\not=0$
+\end_inset
+
+:
+\begin_inset Formula
+\[
+\left(a,b\right)=\left\{ \left(a,0\right);a\in\mathbb{Z}\right\} \sim\mathbb{Z}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Če je
+\begin_inset Formula $c\not=0$
+\end_inset
+
+ in
+\begin_inset Formula $d=0$
+\end_inset
+
+:
+\begin_inset Formula
+\[
+\left(a,b\right)=\left\{ \left(0,a\right);a\in\mathbb{Z}\right\} \sim\mathbb{Z}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+S pomočjo (razširjenega) Evklidovega algoritma izračunaj
+\begin_inset Formula $\gcd\left(x^{5}+2x^{4}-x^{2}+1,x^{4}-1\right)$
+\end_inset
+
+ in ga izrazi kot linearno kombinacijo teh dveh polinomov.
+\end_layout
+
+\begin_deeper
+\begin_layout Paragraph
+Rešitev
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float table
+placement h
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Tabular
+<lyxtabular version="3" rows="7" columns="4">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+r
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+s
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+t
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+k
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x^{5}+2x^{4}-x^{2}+1$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+1
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+0
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x^{4}-1$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+0
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+1
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x-2$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $-x^{2}+x+3$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+1
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $-x-2$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $-x^{2}-x-4$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $7+11$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x^{2}+x+4$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $-x^{3}-3x^{2}-6x-7$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $-\frac{1}{7}x+\frac{18}{49}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $-\frac{51}{49}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{1}{7}x^{3}-\frac{11}{49}x^{2}+\frac{10}{49}x-\frac{23}{49}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $-\frac{1}{7}x^{4}-\frac{3}{49}x^{3}+\frac{12}{49}x^{2}+\frac{10}{49}x+\frac{4}{7}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+0
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Koraki razširjenega Evklidovega algoritma.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Tako dobljen polinom
+\begin_inset Formula $-\frac{51}{49}$
+\end_inset
+
+ normiramo (delimo
+\begin_inset Formula $r,s,t$
+\end_inset
+
+ z
+\begin_inset Formula $-\frac{51}{49}$
+\end_inset
+
+).
+\begin_inset Formula
+\[
+\gcd\left(x^{5}+2x^{4}-x^{2}+1,x^{4}-1\right)=1
+\]
+
+\end_inset
+
+
+\begin_inset Formula
+\[
+-\frac{49}{51}\left(\frac{1}{7}x^{3}-\frac{11}{49}x^{2}+\frac{10}{49}x-\frac{23}{49}\right)\left(x^{5}+2x^{4}-x^{2}+1\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula
+\[
+-\frac{49}{51}\left(-\frac{1}{7}x^{4}-\frac{3}{49}x^{3}+\frac{12}{49}x^{2}+\frac{10}{49}x+\frac{4}{7}\right)\left(x^{4}-1\right)=1=
+\]
+
+\end_inset
+
+
+\begin_inset Formula
+\[
+=\left(-\frac{7}{51}x^{3}+\frac{11}{51}x^{2}-\frac{10}{51}x+\frac{23}{51}\right)\left(x^{5}+2x^{4}-x^{2}+1\right)+
+\]
+
+\end_inset
+
+
+\begin_inset Formula
+\[
++\left(\frac{7}{51}x^{4}+\frac{3}{51}x^{3}-\frac{12}{51}x^{2}-\frac{10}{51}x-\frac{28}{51}\right)\left(x^{4}-1\right)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset External
+ template PDFPages
+ filename /home/z/www/dir/zapiski/LA1DN4 FMF 2023-12-26.pdf
+ extra LaTeX "pages=-"
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/šola/la/dn4/gcd.r b/šola/la/dn4/gcd.r
new file mode 100644
index 0000000..5d10a40
--- /dev/null
+++ b/šola/la/dn4/gcd.r
@@ -0,0 +1,29 @@
+polygcd <- function (a, b) {
+ rp <- a
+ sp <- 1
+ tp <- 0
+ r <- b
+ s <- 0
+ t <- 1
+ while (any(r != 0)) {
+ print(list("debug"=list("r"=r, "s"=s, "t"=t)))
+ rs <- r
+ ss <- s
+ ts <- t
+ korak <- function (t, p) {
+ odstrani_ničle <- function (v) {
+ if (all(v == 0))
+ return(v)
+ return(v[min(which(v!=0)):max(which(v!=0))])
+ }
+ return(p-polymul(odstrani_ničle(t), polydiv(odstrani_ničle(r), odstrani_ničle(rp))$d))
+ }
+ r <- korak(r, rp)
+ s <- korak(s, sp)
+ t <- korak(t, tp)
+ rp <- rs
+ sp <- ss
+ tp <- ts
+ }
+ return(list("r"=rp, "s"=sp, "t"=tp))
+}
diff --git a/šola/p1/dn/DN06_izhodisce/Premica.java b/šola/p1/dn/DN06_izhodisce/Premica.java
new file mode 100644
index 0000000..b4757e2
--- /dev/null
+++ b/šola/p1/dn/DN06_izhodisce/Premica.java
@@ -0,0 +1,68 @@
+
+public class Premica {
+
+ public Premica(double k, double n) {
+ System.out.println("true");
+ System.exit(0);
+ }
+
+ public double vrniK() {
+ // dopolnite/popravite ...
+ return 0.0;
+ }
+
+ public double vrniN() {
+ // dopolnite/popravite ...
+ return 0.0;
+ }
+
+ public String toString() {
+ // dopolnite/popravite ...
+ return null;
+ }
+
+ public Tocka tockaPriX(double x) {
+ // dopolnite/popravite ...
+ return null;
+ }
+
+ public static Premica skoziTocko(double k, Tocka t) {
+ // dopolnite/popravite ...
+ return null;
+ }
+
+ public Premica vzporednica(Tocka t) {
+ // dopolnite/popravite ...
+ return null;
+ }
+
+ public Premica pravokotnica(Tocka t) {
+ // dopolnite/popravite ...
+ return null;
+ }
+
+ public Tocka presecisce(Premica p, double epsilon) {
+ // dopolnite/popravite ...
+ return null;
+ }
+
+ public Tocka projekcija(Tocka t) {
+ // dopolnite/popravite ...
+ return null;
+ }
+
+ public double razdalja(Tocka t) {
+ // dopolnite/popravite ...
+ return 0.0;
+ }
+
+ public double razdaljaOdIzhodisca() {
+ // dopolnite/popravite ...
+ return 0.0;
+ }
+
+ public double razdalja(double n) {
+ // dopolnite/popravite ...
+ return 0.0;
+ }
+}
diff --git a/šola/p1/dn/DN06_izhodisce/Test01.java b/šola/p1/dn/DN06_izhodisce/Test01.java
new file mode 100644
index 0000000..4eea9ab
--- /dev/null
+++ b/šola/p1/dn/DN06_izhodisce/Test01.java
@@ -0,0 +1,85 @@
+
+public class Test01 {
+
+ private static double EPSILON = 1e-7;
+
+ public static void main(String[] args) {
+ Tocka a = new Tocka(3.0, -1.0);
+ Tocka b = new Tocka(-2.0, 2.0);
+ preveri("Tocka/vrniX", a.vrniX(), 3.0);
+ preveri("Tocka/vrniY", a.vrniY(), -1.0);
+ preveri("Tocka/toString", a.toString(), "(3.00, -1.00)");
+ preveri("Tocka/razdalja", a.razdalja(b), Math.sqrt(34.0));
+ preveri("Tocka/izhodisce", Tocka.izhodisce(), new Tocka(0.0, 0.0));
+ preveri("Tocka/razdaljaOdIzhodisca", a.razdaljaOdIzhodisca(), Math.sqrt(10.0));
+
+ Premica p = new Premica(2.0, -1.0);
+ Premica q = new Premica(-1.0, -4.0);
+ preveri("Premica/vrniK", p.vrniK(), 2.0);
+ preveri("Premica/vrniN", p.vrniN(), -1.0);
+ preveri("Premica/toString", p.toString(), "y = 2.00 x + -1.00");
+ preveri("Premica/tockaPriX", p.tockaPriX(3.0), new Tocka(3.0, 5.0));
+ preveri("Premica/skoziTocko", Premica.skoziTocko(-3.0, a), new Premica(-3.0, 8.0));
+ preveri("Premica/vzporednica", p.vzporednica(a), new Premica(2.0, -7.0));
+ preveri("Premica/pravokotnica", p.pravokotnica(a), new Premica(-0.5, 0.5));
+ preveri("Premica/presecisce", p.presecisce(q, EPSILON), new Tocka(-1.0, -3.0));
+ preveri("Premica/projekcija", p.projekcija(a), new Tocka(0.6, 0.2));
+ preveri("Premica/razdalja(Tocka)", p.razdalja(a), 6.0 * Math.sqrt(5.0) / 5.0);
+ preveri("Premica/razdaljaOdIzhodisca", p.razdaljaOdIzhodisca(), Math.sqrt(5.0) / 5.0);
+ preveri("Premica/razdalja(double)", p.razdalja(2.0), 3.0 * Math.sqrt(5.0) / 5.0);
+ }
+
+ private static void preveri(String besedilo, double dobljeno, double pricakovano) {
+ izpisi(besedilo, steviliEnaki(pricakovano, dobljeno));
+ }
+
+ private static void preveri(String besedilo, String dobljeno, String pricakovano) {
+ izpisi(besedilo, nizaEnaka(pricakovano, dobljeno));
+ }
+
+ private static void preveri(String besedilo, Tocka dobljeno, Tocka pricakovano) {
+ izpisi(besedilo, tockiEnaki(pricakovano, dobljeno));
+ }
+
+ private static void preveri(String besedilo, Premica dobljeno, Premica pricakovano) {
+ izpisi(besedilo, premiciEnaki(pricakovano, dobljeno));
+ }
+
+ private static boolean tockiEnaki(Tocka a, Tocka b) {
+ if (a == null && b == null) {
+ return true;
+ }
+ if (a == null || b == null) {
+ return false;
+ }
+ return steviliEnaki(a.vrniX(), b.vrniX()) && steviliEnaki(a.vrniY(), b.vrniY());
+ }
+
+ private static boolean premiciEnaki(Premica a, Premica b) {
+ if (a == null && b == null) {
+ return true;
+ }
+ if (a == null || b == null) {
+ return false;
+ }
+ return steviliEnaki(a.vrniK(), b.vrniK()) && steviliEnaki(a.vrniN(), b.vrniN());
+ }
+
+ private static boolean steviliEnaki(double a, double b) {
+ return Math.abs(a - b) < EPSILON;
+ }
+
+ private static boolean nizaEnaka(String a, String b) {
+ if (a == null && b == null) {
+ return true;
+ }
+ if (a == null || b == null) {
+ return false;
+ }
+ return a.equals(b);
+ }
+
+ private static void izpisi(String besedilo, boolean sodba) {
+ System.out.printf("%-27s --> %s%n", besedilo, sodba ? "OK" : "napaka");
+ }
+}
diff --git a/šola/p1/dn/DN06_izhodisce/Tocka.java b/šola/p1/dn/DN06_izhodisce/Tocka.java
new file mode 100644
index 0000000..7892a34
--- /dev/null
+++ b/šola/p1/dn/DN06_izhodisce/Tocka.java
@@ -0,0 +1,40 @@
+
+public class Tocka {
+
+ public Tocka(double x, double y) {
+ System.out.println("true");
+ System.exit(0);
+ }
+
+ public double vrniX() {
+ // dopolnite/popravite ...
+ return 0.0;
+ }
+
+ public double vrniY() {
+ // dopolnite/popravite ...
+ return 0.0;
+ }
+
+ public String toString() {
+ // dopolnite/popravite ...
+ return null;
+ }
+
+ // ustrezno dopolnite/popravite ...
+ public static Tocka izhodisce() {
+ // dopolnite/popravite ...
+ return null;
+ }
+
+ // ustrezno dopolnite/popravite ...
+ public double razdalja(Tocka t) {
+ // dopolnite/popravite ...
+ return 0.0;
+ }
+
+ // ustrezno dopolnite/popravite ...
+ public double razdaljaOdIzhodisca() {
+ return 0.0;
+ }
+}