09-240/Classnotes for Tuesday September 15

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The real numbers A set $\mathbb {R}$ with two binary operators and two special elements $0,1\in \mathbb {R}$ s.t.

$F1.\quad \forall a,b\in \mathbb {R} ,a+b=b+a{\mbox{ and }}a\cdot b=b\cdot a$

$F2.\quad \forall a,b,c,(a+b)+c=a+(b+c){\mbox{ and }}(a\cdot b)\cdot c=a\cdot (b\cdot c)$

${\mbox{(So for any real numbers }}a_{1},a_{2},...,a_{n},{\mbox{ one can sum them in any order and achieve the same result.}}$

$F3.\quad \forall a,a+0=a{\mbox{ and }}a\cdot 0=0{\mbox{ and }}a\cdot 1=a$

$F4.\quad \forall a,\exists b,a+b=0{\mbox{ and }}\forall a\neq 0,\exists b,a\cdot b=1$

${\mbox{So }}a+(-a)=0{\mbox{ and }}a\cdot a^{-1}=1$

${\mbox{(So }}(a+b)\cdot (a-b)=a^{2}-b^{2})$

$\forall a,\exists x,x\cdot x=a{\mbox{ or }}a+x\cdot x=0$

Note: or means inclusive or in math.

$F5.\quad (a+b)\cdot c=a\cdot c+b\cdot c$

Definition: A field is a set F with two binary operators $\,\!+$: F×F → F, $\times \,\!$: F×F → F and two elements $0,1\in \mathbb {R}$ s.t.

$F1\quad {\mbox{Commutativity }}a+b=b+a{\mbox{ and }}a\cdot b=b\cdot a\forall a,b\in F$

$F2\quad {\mbox{Associativity }}(a+b)+c=a+(b+c){\mbox{ and }}(a\cdot b)\cdot c=a\cdot (b\cdot c)$

$F3\quad a+0=a,a\cdot 1=a$

$F4\quad \forall a,\exists b,a+b=0{\mbox{ and }}\forall a\neq 0,\exists b,a\cdot b=1$

$F5\quad {\mbox{Distributivity }}(a+b)\cdot c=a\cdot c+b\cdot c$

Examples

1. $F=\mathbb {R}$
2. $F=\mathbb {Q}$
3. $\mathbb {C} =\{a+bi:a,b\in \mathbb {R} \}$

   $i={\sqrt {-1}}$

   $\,\!(a+bi)+(c+di)=(a+c)+(b+d)i$

   $\,\!0=0+0i,1=1+0i$

4. $\,\!F_{2}=\{0,1\}$
5. $\,\!F_{7}=\{0,1,2,3,4,5,6\}$
6. $\,\!F_{6}=\{0,1,2,3,4,5\}$ is not a field (counterexample)

Theorem: $\,\!F_{P}$ for $p>1$ is a field IFF $p$ is a prime number

Tedious Theorem

1. $a+b=c+d\Rightarrow a=c$ "cancellation property"
2. $a\cdot b=c\cdot b,(b\neq 0)\Rightarrow a=c$

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