From e95b2ac8aee5c4377d359512ec92bd252cd3ade3 Mon Sep 17 00:00:00 2001 From: Alexander Luzgarev Date: Sun, 16 Oct 2016 16:40:07 +0300 Subject: [PATCH] Fix small typos --- algebra.pdf | Bin 1296552 -> 1296504 bytes set-theory.tex | 8 ++++---- 2 files changed, 4 insertions(+), 4 deletions(-) diff --git a/algebra.pdf b/algebra.pdf index 532161f04ef9ebdf9e86d3224e598548746d922b..dba028abe4aa59604937991cb5e9ecf8f7abe45c 100644 GIT binary patch delta 25508 zcmV(;K-<5lkx%%LPq5gH0yZ<3u}cCee;vtgB)9cGU(rVikZAHfuFfq$93aRcapo|F z$0r*z2oM9AQ|!Ob<60!ESY%19Ne;3!-Q3>3lIFM1H-7u`JKp{NBmJV^d?!#ei{+K9Wr99x zR{E5u=z5A~C>>1-5Po~Os%FaAJmf zba2pwTX3|C(h(W{w-kvg{zpe-f3*~0G!1zo;%J)PO*_T26r5?>QS9>}o{1oM0nI-g zqM3HvhAg7_!vxJzBNMRWI*);X72-bQ*&EWoRm7ea5y{5H^;_Y|2vUAg0lUSMy!b`^ z;wrH7?O7VffQCO42u@}&dqM@`se0y_P;q3SiHig>!?zQ_xc~{tfaIc?fAwj$m7vp` zcFq%Vo#ctDy!3j|6+hoxDQ5d4e?DRe$CTd0j`L$~wM2q4>eJCWG=jTr9pHzM0@G#C zKYvMEc_cAsxJV4jog(rv<8Y1rh}=v<2Q1526PHB%6szDI6Z{yfFOXFYVkWEB!HTSc 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zwawj1SXrmK7GQ-Xtbja~DgG$^ldk^P6=lAO#Q*xDBv#n8op(%0HLZB~!&Yl<0WB!*k^*(^p6ud!H_BYM(x`aob}C=^P}$*H2NO8j5&XFE{< diff --git a/set-theory.tex b/set-theory.tex index 23f1a28..e87cc31 100644 --- a/set-theory.tex +++ b/set-theory.tex @@ -358,7 +358,7 @@ $f\circ g = \id_Y$. Наконец, $g$ называется \end{definition} \begin{lemma}\label{lemma:invertible_left_and_right} -Если у отображение $f\colon X\to Y$ есть левое обратное и правое +Если у отображения $f\colon X\to Y$ есть левое обратное и правое обратное, то они совпадают. Таким образом, отображение обратимо тогда и только тогда, когда оно обратимо слева и обратимо справа. \end{lemma} @@ -649,7 +649,7 @@ $n$. Если $P(0)$ истинно и из истинности $P(0), P(1),\do Отображения $\mb R\times\mb R\to\mb R$, задаваемые формулами $(a,b)\mapsto a+b$, $(a,b)\mapsto ab$, $(a,b)\mapsto a-b$, являются бинарными операциями. Отображение $(a,b)\mapsto a^b$ является бинарной -операцией на множестве $\mb N_{\geq 0}$ положительных натуральных чисел. +операцией на множестве $\mb N$. \end{examples} \begin{definition} @@ -668,7 +668,7 @@ $a,b,c\in X$. Нетрудно видеть, что операции сложения и умножения на множестве вещественных чисел являются ассоциативными и коммутативными, а вот возведение в степень -положительных натуральных положительных чисел не является ни +натуральных чисел не является ни ассоциативной, ни коммутативной операцией. \begin{definition} @@ -791,7 +791,7 @@ $$ (\dots x_1\bullet\dots\bullet x_k\dots) \bullet (\dots x_{k+1}\bullet\dots\bullet x_n\dots). $$ -При этом $1 < k < n$. +При этом $1 \leq k < n$. Предположим сначала, что $k = n-1$. Тогда последняя операция состоит в перемножении скобки, в которой стоят $x_1,\dots,x_{n-1}$, на $x_n$. В