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# Chapter 3 Elementary Number Theory and Methods of Proof PowerPoint Presentation, PPT - DocSlides

pamella-moone | 2019-03-15 | General

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3.5. Direct Proof and Counterexample. 5. Floor & Ceiling. Floor & Ceiling. Definition. Floor. Given any real number . x. , the floor of . x. , denoted âŽ£. x. âŽ¦, is defined as: . âŽ£. x. âŽ¦ = . ID: 756517

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Slide1

Chapter 3

Elementary Number Theory and Methods of Proof

Slide23.5

Direct Proof and Counterexample

5

Floor & Ceiling

Slide3Floor & Ceiling

Definition

Floor

Given any real number

x

, the floor of x, denoted âŽ£xâŽ¦, is defined as: âŽ£xâŽ¦ = n â‡” n â‰¤ x < n + 1.CeilingGiven any real number x, the ceiling of x, denoted âŽ¡xâŽ¤, is defined as: âŽ¡xâŽ¤ = n â‡” n-1 < x â‰¤ n.

Slide4Examples

Compute

âŽ£

x

âŽ¦ and âŽ¡

xâŽ¤ for the following:25/4 âŽ£25/4âŽ¦ = âŽ£6+ 1/4âŽ¦ = 6âŽ¡25/4âŽ¤ = âŽ¡6+ 1/4âŽ¤ = 70.999âŽ£0.999âŽ¦ = âŽ£0 + 999/1000âŽ¦ = 0âŽ¡0.999âŽ¤ = âŽ¡0 + 999/1000âŽ¤ = 1

Slide5Examples

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?

âŽ£1,370/40âŽ¦ = âŽ£34.25âŽ¦ = 34

How many buses will be needed if a partially full bus is allowed?âŽ¡1,370/40âŽ¤ = âŽ¡34.25âŽ¤ = 35

Slide6Addition Property of Floor

Does âŽ£

x

+

y

âŽ¦ = âŽ£xâŽ¦ + âŽ£yâŽ¦?Can you find a counterexample where the case is not true. If so, then you can prove that equality is false.How about x = Â½ and y = Â½ ?âŽ£Â½ + Â½âŽ¦ = âŽ£1âŽ¦ = 1âŽ£Â½âŽ¦ + âŽ£Â½âŽ¦ = 0 + 0 = 0hence, the equality is false.

Slide7Proving Floor Property

Prove that for all real numbers

x

and for all integers

m

, âŽ£x + mâŽ¦ = âŽ£xâŽ¦ + mSuppose x is a particular but arbitrarily chosen real number and m is particular but arbitrarily chosen integer.Show: âŽ£x + mâŽ¦ = âŽ£xâŽ¦ + mLet n = âŽ£xâŽ¦, n is integer n â‰¤

x < n+1n + m

â‰¤

x

+

m

<

n

+

m

+ 1 (add

m

to all sides)

âŽ£

x

+

m

âŽ¦ =

n

+

m

(from previous)

since

n

= âŽ£

x

âŽ¦

Thus

âŽ£

x

+

m

âŽ¦ = âŽ£

x

âŽ¦ +

m

Theorem 3.5.1

Slide8Floor of n/2

Theorem 3.5.2 Floor of n/2

For any

n

, âŽ£n/2âŽ¦ = n/2 (if

n even) or (n-1)/2 (if n odd)ExamplesCompute floor of n/2 for the following:n = 5: âŽ£5/2âŽ¦ = âŽ£2 Â½âŽ¦ = 2 = (5-1)/2 = 2n = 8: âŽ£8/2âŽ¦ = âŽ£4âŽ¦ = 4 = (8)/2 = 4

Slide9Div / Mod and Floor

There is a relationship between div and mod and the floor function.

n

div

d

= âŽ£n / dâŽ¦ n mod d = n â€“ dâŽ£n/dâŽ¦From the quotient-remainder theorem, n = dq + r and 0â‰¤r<d a relationship can be proven between quotient and floor.Theorem 3.5.3If n is any integer and d is a positive integer, and if q = âŽ£n/dâŽ¦ and r

= n â€“ dâŽ£n/dâŽ¦ then,

n

=

dq

+

r

and

0â‰¤r<

d

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