Correct me if I'm wrong, proof problem

[Darya]

Hi! I've been trying to figure out whether I can put "=" sign between two particular sets.

Assume that I have proved that for a function g: R -> R is always true that
g(A ∩ B) ⊆ g(A) ∩ g(B) for all sets A, B ⊆ R.

But it doesn't seem to be true the other way around.

If C ⊆ Z+, D⊆ Z-, consider g(k)={|k|:k€Z}
∃x € C => ∃y€ D: y=-x.
Thus, ∃A={x}, ∃B={y:y=-x}
g(A)=x, g(B)=x. If t€g(A) and t€g(B) => t€ (g(A) ∩ g(B)). Thus, g(A) ∩ g(B)=x.

Similarly, if t€A and t€B => t€A∩B.
With that said, A∩B=∅.
g(A∩B)=g(∅)=∅ => g(A)∩g(D) is not subset of g(A∩B).

Am I mistaken somewhere? Thanks!

 

[Dr.Peterson]

No, you're right that the converse is not true.

Some (in fact, a lot) of your notation is wrong, but what you're saying is that you found a counterexample: If A = {2} [I'm taking a specific example, because we don't need to generalize it to any x], and B = {-2}, and g(x)=|x| for any x∈ℝ, then

• g(A) = {2},
• g{B} = {2},
• g(A ∩ B) = g(∅) = ∅,
• g(A) ∩ g(B) = {2}

so while g(A ∩ B) ⊆ g(A) ∩ g(B), it is not true that g(A) ∩ g(B) ⊆ g(A ∩ B). So you can't say that g(A ∩ B) = g(A) ∩ g(B).

And you were right to consider a function that is not one-to-one in looking for a counterexample, because that is why this happened.

 

[Darya]

No, you're right that the converse is not true.

Some (in fact, a lot) of your notation is wrong, but what you're saying is that you found a counterexample: If A = {2} [I'm taking a specific example, because we don't need to generalize it to any x], and B = {-2}, and g(x)=|x| for any x∈ℝ, then

• g(A) = {2},
• g{B} = {2},
• g(A ∩ B) = g(∅) = ∅,
• g(A) ∩ g(B) = {2}

so while g(A ∩ B) ⊆ g(A) ∩ g(B), it is not true that g(A) ∩ g(B) ⊆ g(A ∩ B). So you can't say that g(A ∩ B) = g(A) ∩ g(B).

And you were right to consider a function that is not one-to-one in looking for a counterexample, because that is why this happened.

Thanks a lot! You are right about my notation being off, I'm working on it )

 

[Dr.Peterson]

Thanks a lot! You are right about my notation being off, I'm working on it )

Which is why I chose not to focus on it! It takes time to get used to it, so you need to work on that and the conceptual content in parallel. You have the right attitude.

Having said that, you might want to ask us how to better say what you said, because there is a lot there that I chose to overlook, that would be worth discussing.