Show that for any I am completely lost. I would really appreciate a detailed exp

Question

Show that for any

I am completely lost. I would really appreciate a detailed explaination so I can apply it to other questions.

Explanation / Answer

Two cases : 1) let a = max {a,b}. a > b = > |a-b| = a-b =>(a+b+|a-b|)/2 = (a+b+a-b)/2 = 2a/2 = a = max {a,b}. 2) let b = max {a,b}. b > a = > |a-b| = b-a =>(a+b+|a-b|)/2 = (a+b+b-a)/2 = 2b/2 = b = max {a,b}. both cases the expression satisfies. Two cases : 1) let a = max {a,b} = > b = min{a,b} a > b = > -b > -a => a = max {a,b} => -b is max{-a,-b} => min{a,b} = -max{-a,-b} = b Similarly, 2) let b = max {a,b} = > a = min{a,b} b > a = > -a > -b => b = max {a,b} => -a is max{-a,-b} => min{a,b} = - max{-a,-b} = a Thus here also both cases satisfy. b + a - max{a,b} = min{a,b} as if a = max{a,b} => b + a - max{a,b} = b = min{a,b} other way also: if b = max{a,b} => b + a - max{a,b} = a = min{a,b} thus we are done. have a nice day !!

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