Subset Relationships between Sets of Functions: U, A, B, C, D, E, F

[Steverino777]

Homework Statement

Let U be the set of all functions having R as domain and range a subset of R:

A={f|f is continuous at each x ∈ R}
B={f|f is differentiable at each x ∈ R}
C={f|f '(x) = 2x + 3 for each x ∈ R}
D={f|f is a quadratic polynomial}
E={f|f(0) = 0}
F={f|f is a linear polynomial}

List all subset relationships between pairs of these 6 sets.

Homework Equations

A is a subset of B when every element of A is also and element of B.

The Attempt at a Solution

I'm having a hard time visualizing all these sets and being able to compare their elements. One thing I know from calc: If f is differentiable; then f is continuous. However, if f is continuous, it doesn't mean that f is necessarily differentiable. So I'm assuming A ⊆ B or B ⊆ A, I just don't know which way it is.

I also know that every set is a subset of itself.

So those are probally the easy answers, I'm having trouble relating the rest of them. If anyone could help me out, that would be great.

 

[EnumaElish]

A is a subset of B when every element of A is also an element of B.

Let f be an element of A. Is f also an element of B?

Let g be an element of B. Is g also an element of A?

For C through F, writing out the equations might help. E.g., what does the "characteristic" element of D look like?

 

[Steverino777]

What would be some elements of a? Would they be like equations? If i knew some examples of elements of each equation then I could determine myself probally which are subsets of which.

 

[ZioX]

Hmm. Enuma's help was sufficient but I will show you how to do one.

Let's investigate E and F. All the elements of E have the property that f(0)=0, so what kind of functions are these? There's lots of them! For example, there's sin(x) and there's 2x. It should be fairly easy to see that any sum or scalar multiple of an element of E is also in E (prove it) so we can generate more elements than you could ever count.

But is E a subset of F? If it was that would mean that every element in E is also in F. We know that sin(x) is in E but it is obviously not in F, as F is the set of functions that can be written as f(x)=ax+b.

Now you have 14 other possibilities to check.

Please write back if you're encountering difficulties.

 

[Steverino777]

Alright what you said, makes a lot of sense. I'm still have some trouble grasping all of this though. I think I know a few answers though, hopefully you can tell me if these are right.

For C f '(x)= 2x+3. That would mean f(x)=x^2 + 3x. This is a qudratic polynomial so I would assume then C ⊆ D. This is also cont. and diff so C ⊆ A and C ⊆ B.

I'm also thinking since all qudratic and linear polynomials (i hope) are cont and diff then, D ⊆ A, D ⊆ B, F ⊆ A, F⊆ B.

Also all diff. functions are cont. so B ⊆ A.

So maybe I'm doing this completely wrong, but I'd appreciate if you could tell me how my answers are. Thank you very much for the help so far.