- #1
mendem03
- 14
- 0
Question
if a < b, then -b < -a
proof
if a < b then a-b<0 and b-a>0
so a-b<0<b-a
so -b<-a
Question
if a<b & c<0, then ab>bc
proof
if ac<ab then ab>bc
then ac<ab>bc
then ac>bc
Question
if a>1 then a^2>a
proof
a*a > 1*a
lemma 1: a*a = a^2
(a*a)*a^-1 = a^2 * a^-1
a*(a*a^-1) = a
a * 1 = a
a = a
so a*a>1*a
a^2>a
if a < b, then -b < -a
proof
if a < b then a-b<0 and b-a>0
so a-b<0<b-a
so -b<-a
Question
if a<b & c<0, then ab>bc
proof
if ac<ab then ab>bc
then ac<ab>bc
then ac>bc
Question
if a>1 then a^2>a
proof
a*a > 1*a
lemma 1: a*a = a^2
(a*a)*a^-1 = a^2 * a^-1
a*(a*a^-1) = a
a * 1 = a
a = a
so a*a>1*a
a^2>a