The Greatest and Least Integer Functions

Question

Does ⌊x⌋ = ⌈x⌉ for all real x? Give reasons for your answer.

Accepted Answer

Welcome back, everyone. If Z is a real number, under what condition does the equation, the floor of Z minus the ceiling of Z equals 0, hold true? A says it's for all real numbers Z. B says only when Z is an integer. C says only when Z is not an integer, and D for no value of Z. Now, if we're going to know under what condition our equation holds true, let's make sense of our equation. Now, the floor of C, this symbol represents the floor function minus the ceiling of Z, this function or this represents the ceiling function, must be equal to 0. Let's let. The flora Z. Be equal to be. And the ceiling of the. OK, the ceiling of Z. Be equal to C. Where B and C are integers. Now, if that's the case, by definition, for our floor function, this means then that Z is going to be greater than or equal to B, but less than B + 1, OK? From when we said the floor of Z equals B. And this also means. That Z must be less than must be greater than C minus 1, but less than or equal to C from when we said the ceiling of Z equals C. No, this equation implies, based on our definitions for the floor of Z and the ceiling of Z, OK, that B minus C is going to be equal to 0, which means that B equals C. If we substitute, substituting B equals C. Into the inequalities then No, they are going to become B less than R equal to Z, less than B + 1 and B minus 1 less than C is less than Z, which is less than or equal to B. Now, how can we make sense of these intervals to figure out what condition will hold true under these two scenarios? Well, let's try to put this on a number line. Let's try to visualize what it's really saying. So, we have B, B + 1 and B minus 1. And if we think about each of these intervals from the first, we know that Z is greater than or equal to B, but less than B + 1. OK. So Z would go from here, OK, to here. And we also know from our second equation or from our second inequality rather that let me highlight this in red here. We also know that Z. Is less than or equal to be less than or equal to be, but greater than B minus 1. So now if we think about the value of Z here from this interval, we can tell at its intersection that Z must be equal to B. No, since B is an integer, that means Z is going to be an integer. Because by definition, we know that Z, like we said earlier, sorry, we know that B, like we said earlier and C are integers. Thus, the equation, the floor of Z minus the ceiling of Z equals 0, holds if and only if Z is an integer. If we look back at our answer choices, that means B is the correct answer. Thanks a lot for watching everyone. I hope this video helps.

The Greatest and Least Integer Functions

Does ⌊x⌋ = ⌈x⌉ for all real x? Give reasons for your answer.

Key Concepts

Greatest Integer Function (Floor Function)

Least Integer Function (Ceiling Function)

Comparison of Floor and Ceiling Functions

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