: For any two functions and from the set ,
: the fact that the function is in the set
: implies that
: is in the set
: and
: is in the set
I'm having trouble understanding this pair of claims:
So g is some function whose result is greater or equal to 0... but I don't really get what the rest of the claim means.
Could someone please explain them to me?
Thanks in advance.
Thanks for the explanation.
So the first claim would be true right? If neither f nor g could be positive, then one of f or g must be smaller than f + g; and if f + g is smaller than c*h(n), then f/g is also smaller than c*h(n)?
And the second would be false? Since either f or g could be greater than f + g if one of them is smaller than 1, then one of them could be greater than c*h(n)?
Ah, now I see. Another reason I was confused is f/g. This problem defines f + g as a function, so, by analogy, I thought that f/g is a function defined by (f/g)(x) = f(x) / g(x). I could not see how division fits the first problem.So the first claim would be true right? If neither f nor g could be negative, then one of f or g must be smaller than f + g; and if f + g is smaller than c*h(n), then f/g is also smaller than c*h(n)?
Finally, it would help to be consistent about the distinction between a function as a whole and its value at a particular point. Namely, in the phrase "f + g is smaller than c*h(n)", f + g is a function and c*h(n) is a number (and what is n)? At the risk of being pedantic, one could write, "if, for all n >= B, (f + g)(n) is smaller than c*h(n), then so are f(n) and g(n)".
Do you mean f * g rather than f + g? Then yes, that makes sense.And the second would be false? Since either f or g could be greater than f + g if one of them is smaller than 1, then one of them could be greater than c*h(n)?
Oh yes I meant "f or g" by the f/g. So basically for the proof all I need to write is something along the lines of "since (f+g)(n) <= c*h(n), and individually f(n) and g(n) <= (f+g)(n) since both could only be positive, then f(n) <= c*h(n) and g(n) <= c*h(n)?
Oh yes I meant f*g. Lol I think I was falling asleep when I wrote that.
Yes, provided it is obvious to you how to relate f(n) <= c*h(n) and -- it is this fact that you need ultimately to show. (This is indeed obvious.)So basically for the proof all I need to write is something along the lines of "since (f+g)(n) <= c*h(n), and individually f(n) and g(n) <= (f+g)(n) since both could only be nonnegative, then f(n) <= c*h(n) and g(n) <= c*h(n)?