SOLUTION: Prove or disprove the statements below. (a) For all positive real numbers x and y, ⌊x*y⌋ ≤ ⌊x⌋*⌊y⌋ (b) For all positive real numbers x

Algebra ->  Proofs -> SOLUTION: Prove or disprove the statements below. (a) For all positive real numbers x and y, ⌊x*y⌋ ≤ ⌊x⌋*⌊y⌋ (b) For all positive real numbers x      Log On


   

Question 1074715: Prove or disprove the statements below.
(a) For all positive real numbers x and y, ⌊x*y⌋ ≤ ⌊x⌋*⌊y⌋
(b) For all positive real numbers x and y, ⌈x*y⌉ ≤ ⌈x⌉*⌈y⌉
I am not sure where to start, could someone please give me some hints on how to solve these kind of question. Thank you in advance.
Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
Counterexample for (a) 

⌊1.6*1.7⌋ ?≤? ⌊1.6⌋*⌊1.7⌋ 
⌊2.72⌋ ?≤? ⌊1⌋*⌊1⌋ 
2 ?≤? 1*1
2 ?≤? 1

False, so (a) is disproved.

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Proof for (b) 

Let i = the integer part of x
Let f = the fraction part of x

Let j = the integer part of y
Let g = the fraction part of y

Then x = i+f, 0 < f < 1
     y = j+g, 0 < g < 1


⌈x*y⌉ ?≤? ⌈x⌉*⌈y⌉
⌈(i+f)*(j+g)⌉ ?≤? ⌈i+f⌉*⌈j+g⌉

⌈ij+ig+fj+fg⌉ ?≤? (i+1)*(j+1)

⌈ij+ig+fj+fg⌉ ?≤? ij+i+j+1

since ig < i and fj < j and fg < 1, that's true.

So the theorem is proved for positive real numbers. 

[it's not true for negative real numbers though.]

Edwin