Question

Let x, y ∈ R. Prove the following:

(a) 0 < 1

(b) For all n ∈ N, if 0 < x < y, then x^n < y^n.

(c) |x · y| = |x| · |y|

Answer 1

Sol.-

Let x ,y R

(a) To prove 0 < 1 we first prove that if a R and a ≠ 0 then a^{​​​​​​2}​​​​​ > 0

Since a R and a ≠ 0 therefore by the Trichotomy Property either a P or - a P

If a P then a^​​​2​​​ = a • a P. [​​​​​By using fact(2) given below]

Also if -a P then a^{​​​​​​2}​​​​​ = (-a)•(-a) P  [by using fact (2) given below]

So we can conclude that if a≠0 then a^{​​​​​​2}​​​​​ >0

Now,

Since 1= 1² it follows that 1²>0

1 > 0 Hence Proved.

(b) Given 0<x<y and x,y R

To prove:- for all nN, x^{​​​​​​n}​​​ < y^{​​​​​​n}

we will prove it by using  Mathematical Induction

For n=1 we have x<y [Given]

let the result hold for n=k i.e.  x^{​​​​​​k}​​​ < y^{​​​​​k} -------(1)

Now we will prove that the result is also hold for n= k+1

So for n= k+1

We have x^{​​​​​k+1} = x^{​​​​​​k}​​​​​ • x < y^{​​​​​k} • y [ by using induction hypothesis (1) and x<y is given]

   x^{​​​​​k+1} < y^{​​​​​​k+1}

thus result holds for n= k+1

By using Mathematical Induction

  x^{​​​​​​n} < y^{​​​​​​n} for all n N. Hence Proved.

(c) let x , y R

​​​​​​To prove:- | x•y| = |x|•|y|

if either x or y is 0 then both sides are equal to 0. So we are done here.

Now if neither x nor y is 0 then there are four cases.

case(i) if x>0 ,y>0 then x•y> 0

So, |x•y| =x•y = |x|•|y| [ since x>0,y>0 so |x|=x and |y|=y]

|x•y|=|x|•|y|.

Case(ii) if x>0 ,y<0 then x•y<0

So, |x•y|= - x•y = x•(-y) = |x|•|y| [as x>0 ,y<0 so |x|=x and |y|=-y]

|x•y|=|x|•|y|

Case(iii) if x<0 ,y>0 then x•y <0

So, |x•y|= -x •y = |x|•|y|. [As x<0 ,y>0 so |x|= -x and |y|=y]

|x•y|=|x|•|y|

Case(iv) if x<0 ,y<0 the x•y >0

So, |x•y|= x•y = (-x)•(-y) = |x|•|y| [as x<0,y<0 so |x|=-x and |y|=-y]

|x•y|=|x|•|y|

Hence, | x•y|=|x|•|y| Hence Proved.

Some Facts used here:-

(1) Trichotomy Property:- if a belongs to R then exactly one of the following holds-

a P , a=0 , -a P  

Where P is the set of positive real numbers.

(2) if a,b belongs to P then ab P

(3) Definition of the absolute value:-

The absolute value of a real number x denoted by |x| , is defined by