Mathematical Preliminaries for COMP372 Algorithms

The floor of \(x\) is written \( \lfloor x \rfloor \) and takes real value \(x\) and returns the greatest integer \(\leq x \). For example \( \lfloor 5.4 \rfloor \) = 5, \( \lfloor -3.2 \rfloor \) = -4, \( \lfloor 5 \rfloor \) = 5. The ceiling of \(x \) is written \( \lceil x \rceil \) and takes integer \(\geq x \). For example, \( \lceil 5.4 \rceil \) = 6, \( \lceil -3.2 \rceil \) = -3, \( \lceil 5 \rceil \) =5.

1. Which of the following is untrue? \(x\) is a real number and \(n\) is an integer.

\( \lfloor x +y \rfloor\) = \( \lfloor x \rfloor + \lfloor y \rfloor\) \( \lceil x \rceil \) < \(x\) +1 \( \lfloor x+n \rfloor \) = \( \lfloor x \rfloor \) +\(n\) \( \lceil x+n \rceil \) = \( \lceil x \rceil \) +\(n\)

2. Which of the following is untrue? \(x\) is a real number and \(n\) is an integer.

\( \lfloor n/2 \rfloor + \lceil n/2 \rceil \) = n \( \lceil lg(n+1) \rceil \) = \( \lfloor lgn \rfloor \) + 1 \( \lfloor n/2 \rfloor + \lfloor n/2 \rfloor = n \) \( \lceil x \rceil -1 < x \)

3. If \(X = \lfloor X \rfloor = \lceil X \rceil \) then

\( X \) is a fractional number \( X \) is a Integer \( X \) is less than 1 none of the mentioned

4. \(\lfloor 2.4 \rfloor + \lceil 2.9 \rceil \) is equal to ______

4 6 5 none of the mentioned Submit

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