I have a set consisting of $0$ and $1$ with addition and multiplication as operations. $$0+0=1+1=0,\ \ \ \ \ \ \ \ \ \ 1+0=0+1=1$$ $$0 \cdot 0= 0\cdot 1=1\cdot 0=0, \ \ \ \ \ \ \ \ \ \ 1 \cdot 1=1$$
Now I have to verify that it is a field. So the properties of a field is:
a) $a+b=b+a$ and $a\cdot b=b \cdot a$.
$1+0=0+1=1$ and $ 0\cdot 1=1\cdot 0=0$
b) $(a+b)+c=a+(b+c)$ and $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
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c) $a(b+c)=ab+ac$
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d) There are distinct real numbers $0$ and $1$ such that $a+0=a$ and $a\cdot1=a$ $\forall a$.
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e) For each $a$ there is a real number $-a$ such that $a+(-a)=0$ , and if $a\ne 0$, there is a real number $1/a$ such that $a(1/a)=1$.
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My question is how does the set fulfil b), c), d), and e)?