2010 March | TICS

Archive for March, 2010

Saturday, March 27th, 2010

Proposed in Mathematics: A Very Short Introduction. Given the following rules:

A1 Commutative law for addition: $\forall a,b\quad a + b = b + a$

A2 Associative law for addition: $\forall a,b,c\quad a + (b + c) = (a + b) + c$

A3 Additive identity: $\forall a\quad 0 + a = a$

A4 Additive inverse: $\forall a,\exists b\quad a + b = 0$

A5 Cancellation law for addition: $\forall a,b,c\quad if\ a + b = a + c,\ then\ b = c$

M1 Commutative law for multiplication: $\forall a,b\quad ab = ba$

M2 Associative law for multiplication: $\forall a,b,c\quad a(bc) = (ab)c$

M3 Multiplicative identity: $\forall a\quad 1a = a$

M4 Multiplicative inverse: $\forall a\ne0,\ \exists b\quad ab = 1$

M5 Cancellation law for multiplication:

$\forall a,b,c\quad if\ a\ne0\ and\ ab = ac,\ then\ b = c$

D Distributive law: $\forall a,b,c\quad (a + b)c = ac + bc$

Let’s prove that (-1)(-1) = 1: