Upper bound and lower bound

That means that ln(n) is a lower bound for ln(3n) and so f(n)= ln(2n) is a lower bound for g(n)= ln(3n). In fact, we can say more than that. ln(3n)= ln(n)+ ln(3) so ln(3n)- ln(n)= ln(3)> 0 for all n> 0. That means that ln(3n)> ln(n) for all n> 0 so ln(3n) is an upper bound for ln(n). That means that f(n)= ln(2n) is an upper bound for g(n)= ln(n).
  • #1
l46kok
297
0

Homework Statement



1.
f(n) = n - 100
g(n) = n - 200

2.
f(n) = log(2n)
g(n) = log(3n)

n >= 0 in all cases
Find out if f(n) is an upperbound, lowerbound or both of g(n)

Homework Equations





The Attempt at a Solution



in case of 1, f(n) has to be an upperbound of g(n) because when graphed together, f(n) has to be an upperbound of g(n).

For 2, solution does not exist at n = 0. Otherwise, f(n) is a lower bound of g(n). Does this mean that f(n) is a lower bound or both?
 
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  • #2
l46kok said:

Homework Statement



1.
f(n) = n - 100
g(n) = n - 200

2.
f(n) = log(2n)
g(n) = log(3n)

n >= 0 in all cases
Find out if f(n) is an upperbound, lowerbound or both of g(n)

Homework Equations





The Attempt at a Solution



in case of 1, f(n) has to be an upperbound of g(n) because when graphed together, f(n) has to be an upperbound of g(n).
Too vague. What you should say is "200> 100 so -100> -200 and n- 100> n- 200 for all n. Since f(n)> g(n) for all n, f is an upper bound of g."

For 2, solution does not exist at n = 0. Otherwise, f(n) is a lower bound of g(n). Does this mean that f(n) is a lower bound or both?
Is suspect that should not be "[itex]n\ge 0[/itex] for both cases" but only n> 0 for the second. As you point out, ln(0) is not defined so the problem makes no sense for n= 0.

ln(2)< ln(3) so ln(n)+ ln(2)< ln(n)+ ln(3) for all n> 0. ln(2n)< ln(3n) for all n> 0.
 

1. What is an upper bound and lower bound in mathematics?

An upper bound and lower bound are mathematical concepts used to describe the maximum and minimum possible values of a set of numbers or a function. The upper bound is the highest possible value, while the lower bound is the lowest possible value.

2. How are upper and lower bounds used in data analysis?

In data analysis, upper and lower bounds are used to set limits on the possible outcomes of a study or experiment. This helps to ensure that the results are within a certain range and are not influenced by extreme outliers.

3. Can upper and lower bounds be the same value?

Yes, in some cases, the upper and lower bounds can be the same value. This means that the set of numbers or function has a single possible outcome, rather than a range of possible outcomes.

4. How do you calculate the upper and lower bounds of a data set?

The upper and lower bounds can be calculated by finding the maximum and minimum values in the data set. Alternatively, they can also be calculated using statistical methods such as standard deviation or confidence intervals.

5. What is the difference between an absolute and relative upper and lower bound?

An absolute upper or lower bound is a fixed value that cannot be exceeded or decreased, while a relative upper or lower bound is a percentage or proportion of the maximum or minimum value. Relative bounds are useful in situations where the data set is large and the values are expected to change over time.

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