Now using just the axioms 112 as numbered in text or 14 as n

Now using just the axioms 1-12 (as numbered in text) or 1-4 (as numbered in class), prove the following. For x, y R, if x lessthanorequalto y and y lessthanorequalto x, then x = y. For x, y R, if xy > 0 then either x > 0 and y > 0, or x

Solution

a)

x <= y , So (y-x) >=0

y <= x So (x--y) >=0

multiplying both

-(y-x)^2 >=0

square is positve

so y-x=0

y=x

b) xy>0

so if exactly oneis positive then xy cannot be greater than zero

suppose only one is positive

x> 0, but y<0

y < 0

multpliying both sides by x (equality doesn\'t change with multiplication of positive number)

so xy < 0

c ) (-1).(-1) + (-1).(1) = (-1) ( 1 + -1) = 0

so ( -1).(-1) = - (-1).(1)

and similarly (-1).(1) +(1).(1) = 0, so -(-1).(1) = (1).(1)

since additive inverse is unique ( -1).(-1) =(1) .(1) =1

d) if x>=0

then x>= 0

multiply

x^2 >=0

if x<=0

-x >=0

-x >=0

multiply

(-x).(-x) = (-1).(-1)x^2 =x^2 >=0