Greatest Integer Function

Algebra Level 3

For a real number r r , [ r ] \left[r\right] denotes the largest integer less than or equal to r r .If x , y x,y are real numbers with x , y ≥ 1 x,y\geq 1 then which of the following statement is always true?

Source : KVPY 2014
[ 2 x ] ≤ 2 [ x ] [2^x]\leq 2^{[x]} [ x y ] ≤ [ x y ] \left [\frac{x}{y}\right ]\leq\left [\frac{x}{y}\right ] [ x y ] ≤ [ x ] [ y ] [x y]\leq[x][y] [ x + y ] ≤ [ x ] + [ y ] [x+y]\leq [x]+[y]

Akshay Sharma by Akshay Sharma , 21, India

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1 solution

Sharky Kesa
May 1, 2016

Obviously, ⌊ x y ⌋ ≤ ⌊ x y ⌋ \left \lfloor \dfrac{x}{y} \right \rfloor \leq \left \lfloor \dfrac{x}{y} \right \rfloor since they're the same value!

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How can a number be greater than or equal to itself

Sid Rana - 5 years, 1 month ago

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Since they are equal, and equality is a subset of being greater than or equal to so a number can be greater than or equal to itself.

Sharky Kesa - 5 years, 1 month ago

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Understood. Thanks!

Sid Rana - 5 years, 1 month ago

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