# Thread: Analysis, functions. HELP.

1. ## Analysis, functions. HELP.

Could someone prove the next statement formally?

2. ## Re: Analysis, functions. HELP.

Originally Posted by Overlord265
Could someone prove the next statement formally?
This is known as a definitional proof. That is the proof just "falls out of the definitiion.
If $t\in f(A)$ then $(\exists a_t\in A)[f(a_t)=t$.
BUT $A\subseteq B$ so $a_t\in B$ WHY?]
Does tha mean that $t\in f(B)~?$ EXPLAIN!

3. ## Re: Analysis, functions. HELP.

Assume that there is a mapping from X to some Y, and a partial function from B to Y. Using the definition of a subset, which states that for every a from A , a belongs to B, I can further assume that set B can be expressed as a union of set A and difference of sets A and B. Now substituting B for the sum in the earlier defined partial function, we can see that the original statement is true in both cases whether A is a proper subset or not. Anyway, I doubt it can be considered as the solid proof.

4. ## Re: Analysis, functions. HELP.

Originally Posted by Overlord265
Assume that there is a mapping from X to some Y, and a partial function from B to Y. Using the definition of a subset, which states that for every a from A , a belongs to B, I can further assume that set B can be expressed as a union of set A and difference of sets A and B. Now substituting B for the sum in the earlier defined partial function, we can see that the original statement is true in both cases whether A is a proper subset or not. Anyway, I doubt it can be considered as the solid proof.
What in the world is a partial function? I have never seen such.