*** empty log message ***

ross-3.1.tex

@@ -0,0 +1,43 @@
+\beginsection{3.1}
+
+(a) Which of the properties A1--A4, M1--M4, DL, O1--O5 fail for the
+natural numbers $N$?
+
+\medskip
+A1. $a+(b+c)=(a+b+c)$
+
+A2. $a+b=b+a$
+
+A3. $a+0=a$
+
+A4. For each $a$ there is as element $-a$ such that $a+(-a)=0$.
+{\it Fails, no negative numbers.}
+
+M1. $a(bc)=(ab)c$
+
+M2. $ab=ba$
+
+M3. $a\cdot1=a$
+
+M4. For each $a\ne0$ there is an element $a^{-1}$ such that $aa^{-1}=1$.
+{\it Fails, no multiplicative inverse.}
+
+DL. $a(b+c)=ab+ac$
+
+O1. Either $a\le b$ or $b\le a$.
+
+O2. If $a\le b$ and $b\le a$ then $a=b$.
+
+O3. If $a\le b$ and $b\le c$ then $a\le c$.
+
+O4. If $a\le b$ then $a+c\le b+c$.
+
+O5. If $a\le b$ and $0\le c$ then $ac\le bc$.
+{\it Interesting since $0\not\in N$.}
+
+\medskip
+(b) Which of these properties fail for the set of integers $Z$?
+
+\medskip
+M4, no multiplicative inverse.
+

ross.tex

@@ -13,6 +13,8 @@
 \input ross-2.2.tex
 \input ross-2.3.tex
 
+\input ross-3.1.tex
+
 \input ross-7.1.tex
 \input ross-7.2.tex