Prove the following i IxyI IxIIyISolutioni Case 1xy0 xyxy C

Prove the following:

(i) IxyI = IxI*IyI.

Solution

i)

Case 1:xy>=0
|xy|=xy

Case 1.1 x>=0,y>=0

x=|x|,y=|y|,|xy|=|x||y|

Case 1.2 x<0,y<0

x=-|x|,y=-|y|

xy=|x||y|

Hence, |xy|=|x||y|

Case 2:xy<0

|xy|=-xy

Case 2.1 x>0,y<0

   x=|x|, y=-|y|

xy=-|x||y| ,-xy=|x||y|

Case 2.2 x<0,y>0

x=-|x|,y=|y|

xy=-|x||y| , -xy=|x||y|

Hence, |xy|=|x||y|

Hence proved

ii)

Case 1.x-y>=0

|x-y|=x-y =x+(-y)<=|x|+|y|

Case 2. x-y<0

|x-y|=-(x-y)=-x+y<=|x|+|y|

Hence proved

iii)

Case 1.x-y>=0

|x-y|=x-y =x-y>=|x|-|y|

Case 2. x-y<0

|x-y|=-(x-y)=-x+y>=|x|-|y|

Hence proved

iv)

Using iii)

||x|-|y||<=||x-y||=|x-y|

Hence proved