# Mathematical Reasoning and Writing - Counterexamples with subsets.

1. Apr 1, 2013

### mliuzzolino

1. The problem statement, all variables and given/known data

Let f: A --> B be a function and let S, T $\subseteq$ A and U, V $\subseteq$ B.

Give a counterexample to the statement: If f (S) $\subseteq$ f (T); then S $\subseteq$ T:

2. Relevant equations

3. The attempt at a solution

PF:

Assume f(S) $\subseteq$ f(T).

Let x $\in$ S.

Then $\exists$ y $\in$ f(S) $\ni$ f(x) = y.

Since f(S) $\subseteq$ f(T), y $\in$ f(T).

****

Suppose $\forall$ a $\in$ T where a ≠ x, $\exists$ y $\in$ f(T) $\ni$ f(a) = y.

Then x $\notin$ T.

Q.E.D.

I am not exactly sure I am doing this right, especially the reasoning beyond the ****. I almost have the feeling I should use the pre image of f(T) somehow to show that x $\notin$ T.

Why can I not just say that $\exists$ x $\in$ S where x $\notin$ T? Would that not suffice as a counterexample in such a general proof as this?

2. Apr 1, 2013

### Dick

You aren't supposed to do a general proof. The statement isn't false for all functions, only some. You have to think of one.

3. Apr 1, 2013

### mliuzzolino

Oh! I don't know why I was thinking what I was.

Proof:
Let f: ℝ → [0, ∞) by f(x) = x2.
Q.E.D.

The negative ℝ could be considered S and the positive ℝ could be considered T. Then by f(x) = x2, f(S) is contained in f(T), but obviously S is not contained in T.

Would this be a suitable counterexample?

4. Apr 1, 2013

### micromass

Staff Emeritus
That's perfect!

5. Apr 1, 2013

### micromass

Staff Emeritus
Some comments on your OP. When writing a proof, you should never use the symbols $\exists$ and $\forall$. You should always write it out in words. This is a very common mistake that new people make and it's one way I see whether somebody is used to proving things or not.

And your proof should be much wordier. A proof should really be read like an english text.

Last edited: Apr 1, 2013
6. Apr 1, 2013

### Curious3141

I had no idea walruses observe the 1st of April.

7. Apr 1, 2013

### micromass

Staff Emeritus
I was serious

8. Apr 1, 2013

### Curious3141

Haha, really?

Seriously, I do find a proof with all those logical operators a real PITA to read. I'd much prefer they wrote it all in words. But then, I'm not a mathematician, your walrus-ness. :tongue2: