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(a) Which of the properties A1--A4, M1--M4, DL, O1--O5 fail for the
natural numbers $N$?

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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 $0\not\in N$.}

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(b) Which of these properties fail for the set of integers $Z$?

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M4, no multiplicative inverse.

