Definition maxx y largest number between x and y so max3 5

Definition: max{x, y} = largest number between x and y. so max{3, 5} = 5 the same for mm{x, y} Lot x and y 1be real numbers. Prove the following using proof by cases. max {x, y} = x + y + |x - y|/2 max {x, y} + min {x, y} = x + y

Solution

a)

Case 1:x>=y

max{x,y}=x

(x+y+|x-y|)/2=(x+y+x-y)=x

Case 2:x<y

max{x,y}=y

(x+y+|x-y|)/2=(x+y-(x-y))/2=y

HEnce proved

b)

Case 1: x>=y

max{x,y}+min{x,y}=x+y

Case 2:x<y

max{x,y}+min{x,y}=y+x=x+y

Hence proved