SAT1000 - P866

Algebra Level 3

For all real numbers $x$ and $y$ , which of the following options is always true?

$A.\ \lfloor -x \rfloor=-\lfloor x \rfloor$

$B.\ \lfloor 2x \rfloor=2\lfloor x \rfloor$

$C.\ \lfloor x+y \rfloor \leq \lfloor x \rfloor + \lfloor y \rfloor$

$D.\ \lfloor x-y \rfloor \leq \lfloor x \rfloor - \lfloor y \rfloor$

Have a look at my problem set: SAT 1000 problems

B

C

D

A

1 solution

Alexander McDowell
Jun 19, 2020

Option A cannot be true because if $x = 0.5$ , then $\lfloor -0.5 \rfloor = -1$ and $-\lfloor 0.5 \rfloor = 0$ . The two are not equal to each other, therefore Option A is not true.

Option B is also false because if $x = 0.5$ , then $\lfloor 2 * 0.5 \rfloor = 1$ and $2\lfloor 0.5 \rfloor = 0$ . The two are not equal to each other, therefore Option B is not true.

Option C is also false because if $x = 0.5$ and $x = 0.6$ , then $\lfloor 0.5 + 0.6 \rfloor = 1$ and $\lfloor 0.5 \rfloor + \lfloor 0.6 \rfloor = 0$ . The two are not equal to each other, therefore Option B is not true.

By the process of elimination, Option $\boxed{D}$ is the correct answer.

@Alexander McDowell You copied the line of option B for Option C. It should have been 1 > 0 hence option C is not true.

Siddharth Chakravarty - 11 months, 3 weeks ago

Tried that maybe some more research is needed

Shubham Singh Chahar - 11 months, 3 weeks ago

Maybe one more nested comment here

Shubham Singh Chahar - 11 months, 3 weeks ago

@Shubham Singh Chahar – Or more more comment here

Shubham Singh Chahar - 11 months, 3 weeks ago

@Shubham Singh Chahar – And one more after it, lets see where this app will go

Shubham Singh Chahar - 11 months, 3 weeks ago

I cannot figure out what is the least number of words I should type for this app to allow me to comment

Shubham Singh Chahar - 11 months, 3 weeks ago

What is this symbol "⌊ ⌋" ? I thought it was absolute value and got it wrong.

A Former Brilliant Member - 11 months, 3 weeks ago

It is called floor down. It is round down the value i.e to make it the value of the smallest integer to it. Like ⌊2.98⌋ = 2 and ⌊-1.2⌋ = -2 as -2 < -1

Siddharth Chakravarty - 11 months, 3 weeks ago

It is not necessary that if all rest of the options are incorrect then the last answer must be correct, it can happen that the question itself is wrong. Give full solution (show why $D$ is not incorrect )

Zakir Husain - 11 months, 2 weeks ago

Explanation for Option D: Consider three cases, one in which x > y, one in which y > x, and one where x = y.

Case 1: x > y. Let x = y + n + m where n is a positive integer and m is a real-valued number such that 0 < m < 1. Thus, floor(y + n + m - y) <= floor(y + n + m) - floor(y) leads to n <= y + n - y (the m is between zero and one so it rounds down) or n <= n which is true. D therefore holds for x > y.

Case 2: x < y. Now let y = x + n + m with the same rules for n and m. Thus, floor(x - x - n - m) <= floor(x) - floor(x + n + m) leads to -n - 1 <= x - x - n (the m is between -1 and 0 so it rounds down to -1) or -n - 1 <= -n which is true. D therefore holds for x < y.

Case 3: x = y. It is clear that if x = y then the inequality becomes 0 <= 0 which is true.

Since we have shown that A, B, and C are not true for all real numbers x and y, D is the only option among the choices which is valid.

Alexander McDowell - 11 months, 2 weeks ago

SAT1000 - P866

1 solution

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