   Chapter 11.2, Problem 17ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Use the definition of Θ -notation to show that ⌊ n 2 ⌋ is Θ ( n ) . (Hint: Show that if n ≥ 4 , then n 2 − 1 ≥ 1 4 n .)

To determine

To show:

That n2 is Θ(n) using the definition of Θ notation.

Explanation

Given information:

If n4 then, n2114n is given as a hint.

Formula used:

Let f and g be real valued functions defined on the same nonnegative integers, with g(n)0 for every integer nr, where r is positive real number.

Then,

f is of order g, written f(n) is Θ(g(n)), if and only if, there exist positive real numbers A,B and kr such that

Ag(n)f(n)Bg(n) for every integer nk.

Proof:

Let f(n)=n2 and g(n)=n, for any integer n4.

Hence,

For all n0, n+12>0 and n4

Then, n20 and n4

Obviously, f(n)0 and g(n)0

For all n4 we can show that,

n4n41n4+n41+n4

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