Floor and Ceiling - Discrete Mathematical Structures - Lecture Slides, Slides of Discrete Mathematics

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Chapter 3

Elementary Number Theory and

Methods of Proof

Direct Proof and Counterexample 5

Floor & Ceiling

Examples

• Compute ⎣x⎦ and ⎡x⎤ for the following:

• 25/
  • ⎣25/4⎦ = ⎣6+ 1/4⎦ = 6
  • ⎡25/4⎤ = ⎡6+ 1/4⎤ = 7
• • ⎣0.999⎦ = ⎣0 + 999/1000⎦ = 0
  • ⎡0.999⎤ = ⎡0 + 999/1000⎤ = 1

Examples

• The 1,370 soldiers at a military base a re given

the opportunity to take buses into town for an

evening out. Each bus holds a maximum of 40

passengers

• What is the maximum number of buses the base

will send if only full buses are sent?

• How many buses will be needed if a partially full

bus is allowed?

Proving Floor Property

• Prove that for all real numbers x and for all

integers m, ⎣x + m⎦ = ⎣x⎦ + m

• Suppose x is a particular but arbitrarily chosen real

number and m is particular but arbitrarily chosen

integer.

• Show: ⎣x + m⎦ = ⎣x⎦ + m
  • Let n = ⎣x⎦, n is integer n ≤ x < n+
  • n + m ≤ x + m < n + m + 1 (add m to all sides)
  • ⎣x + m⎦ = n + m (from previous)
  • since n = ⎣x⎦
  • Thus ⎣x + m⎦ = ⎣x⎦ + m

• Theorem 3.5.

Floor of n/

• Theorem 3.5.2 Floor of n/

• For any n, ⎣n/2⎦ = n/2 (if n even) or (n-1)/2 (if n

odd)

• Examples

• Compute floor of n/2 for the following:
  • n = 5: ⎣5/2⎦ = ⎣2 ½⎦ = 2 = (5-1)/2 = 2
  • n = 8: ⎣8/2⎦ = ⎣ 4 ⎦ = 4 = (8)/2 = 4