MHB Are These Statements About Preimages Correct and Complete?

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The discussion focuses on verifying the correctness of several statements regarding preimages in set theory. The first three statements about unions, complements, and set differences are confirmed to be accurate through logical proofs. The fourth statement regarding the intersection of preimages is also validated, showing that if two sets are disjoint, their preimages must be disjoint as well. However, the fifth statement, which suggests that if one set is a subset of another, the preimages maintain that relationship, is shown to be false, particularly when considering specific cases like empty sets. Overall, the participants engage in clarifying the implications of these statements and their proofs.
mathmari
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Hey! :o

Let $f:M\rightarrow N$, $A,B\subset N$. I want to check if the following statements about the preimages are correct:
  1. $f^{-1}(A\cup B)=f^{-1}(A)\cup f^{-1}(B)$
  2. $f^{-1}(\overline{A})=\overline{f^{-1}(A)}$
  3. $f^{-1}(A\setminus B)=f^{-1}(A)\setminus f^{-1}(B)$
  4. $A\cap B=\emptyset \Rightarrow f^{-1}(A)\cap f^{-1}(B)=\emptyset$
  5. $A\subseteq B \Rightarrow f^{-1}(B)\subseteq f^{-1}(A)$
I have done the following:

  1. Let $x\in f^{-1}(A\cup B)$. Then it follows that $f(x)\in A\cup B$. This means that either $f(x)\in A$ or $f(x)\in B$, so $x\in f^{-1}(A)$ or $x\in f^{-1}(B)$. It follows that $x\in f^{-1}(A)\cup f^{-1}(B)$.

    We have shown that $\displaystyle{f^{-1}(A\cup B)\subseteq f^{-1}(A)\cup f^{-1}(B)}$. Let $x\in f^{-1}(A)\cup f^{-1}(B)$. Then we either have that $x\in f^{-1}(A)$ or $x\in f^{-1}(B)$, so either $f(x)\in A$ or $f(x)\in B$. This means that $f(x)\in A\cup B$. It follows that $x\in f^{-1}(A\cup B)$.

    We have shown that $\displaystyle{f^{-1}(A)\cup f^{-1}(B) \subseteq f^{-1}(A\cup B)}$. From these two inclusions it follows that $\displaystyle{f^{-1}(A\cup B)=f^{-1}(A)\cup f^{-1}(B)}$.

    $$$$
  2. $$x\in f^{-1}(\overline{A}) \iff f(x) \in \overline{A} \iff f(x) \notin A \iff x\notin f^{-1}(A) \iff x\in \overline{f^{-1}(A)}$$
    Therefore, we get that $f^{-1}(\overline{A})=\overline{f^{-1}(A)}$.

    $$$$
  3. $$x\in f^{-1}(A\setminus B) \iff f(x)\in A\setminus B \iff f(x) \in A \text{ AND } f(x) \notin B \iff x\in f^{-1}(A) \text{ AND } x\notin f^{-1}(B) \\ \iff x\in f^{-1}(A)\setminus f^{-1}(B)$$
    Therefore, we get that $f^{-1}(A\setminus B)=f^{-1}(A)\setminus f^{-1}(B)$.

    $$$$
  4. We suppose that $f^{-1}(A)\cap f^{-1}(B)\neq\emptyset$. That means that $\exists x\in f^{-1}(A)\cap f^{-1}(B)$. That means that $x\in f^{-1}(A)$ AND $x\in f^{-1}(B)$. That implies that $f(x)\in A$ AND $f(x)\in B$, and so $f(x)\in A\cap B$. We get a contradiction since $A\cap B=\emptyset$.
    So, that what we supposed at the beginning was wrong, i.e., $f^{-1}(A)\cap f^{-1}(B)\neq\emptyset$ is not correct. So, it must be $f^{-1}(A)\cap f^{-1}(B)=\emptyset$.
Is everything correct and are the proofs complete? Or could I improve something?

Could you give me a hint about the last statement?

(Wondering)
 
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mathmari said:
Could you give me a hint about the last statement?
(Wondering)
The empty set is a subset of any other set.

I'm sorry that I do not have the time to check your other work, but this is at least a hint about the last statement.
 
Krylov said:
The empty set is a subset of any other set.
mathmari said:
5. $A\subseteq B \Rightarrow f^{-1}(B)\subseteq f^{-1}(A)$
So do you mean to take $A=\emptyset$ ? (Wondering)
 
mathmari said:
So do you mean to take $A=\emptyset$ ? (Wondering)
Yes, and $B = N$.
What do you think about the veracity of statement #5?
 
Hey mathmari! (Smile)

You wrote: '$f(x)∈A∪B$. This means that either $f(x)∈A$ or $f(x)∈B$.'

Using 'either ... or ...' means that they cannot both be true, but that's not the case is it?
Shouldn't it be: '$f(x)∈A∪B$. This means that $f(x)∈A$ or $f(x)∈B$.'? (Wondering)
I think all occurrences of 'either' should be removed.

Other than that I think everything is correct! (Nod)
 
Krylov said:
Yes, and $B = N$.
What do you think about the veracity of statement #5?

So, for $A=\emptyset$ and $B=N$ we have that $\emptyset \subseteq N$.
Let $x\in f^{-1}(N)$ then we have that $f(x)\in N$.
It cannot hold that $f(x)\in \emptyset$ and so it cannot hold that $x\in f^{-1}(\emptyset)$.
Therefore, we have that $f^{-1}(N)\nsubseteq f^{-1}(\emptyset)$.
And so generally the implication $A\subseteq B \Rightarrow f^{-1}(B)\subseteq f^{-1}(A)$ does not hold.

Is this correct? (Wondering)
I like Serena said:
You wrote: '$f(x)∈A∪B$. This means that either $f(x)∈A$ or $f(x)∈B$.'
Using 'either ... or ...' means that they cannot both be true, but that's not the case is it?
Shouldn't it be: '$f(x)∈A∪B$. This means that $f(x)∈A$ or $f(x)∈B$.'? (Wondering)
I think all occurrences of 'either' should be removed.
Other than that I think everything is correct! (Nod)

Ah ok! (Smile)

So, by '$f(x)∈A∪B$. This means that $f(x)∈A$ or $f(x)∈B$.' it is meant that both can be true, isn't it? Or do we have to mention it? (Wondering)
 
mathmari said:
Is this correct?

So, by '$f(x)∈A∪B$. This means that $f(x)∈A$ or $f(x)∈B$.' it is meant that both can be true, isn't it? Or do we have to mention it?

Yep. All correct.
And there is no need to mention that both can be true - that is implied in the word 'or'.
If we want to be explicit about it, we can use 'and/or'. (Mmm)
 
I like Serena said:
Yep. All correct.
And there is no need to mention that both can be true - that is implied in the word 'or'.
If we want to be explicit about it, we can use 'and/or'. (Mmm)

Great! (Mmm)

Is the implication $A\subseteq B \Rightarrow f^{-1}(B)\subseteq f^{-1}(A)$ true when a specific condition holds or is this never true? (Wondering)
 
mathmari said:
Is the implication $A\subseteq B \Rightarrow f^{-1}(B)\subseteq f^{-1}(A)$ true when a specific condition holds or is this never true? (Wondering)

Can we find an example that it's true?
What happens if $A$ and $B$ are $\varnothing$, equal to each other, or $N$? (Wondering)
 
  • #10
I like Serena said:
Can we find an example that it's true?
What happens if $A$ and $B$ are $\varnothing$, equal to each other, or $N$? (Wondering)

Ah if $A=B$ we have that $A\subseteq B=A$ and that $f^{-1}(B)=f^{-1}(A)\subseteq f^{-1}(A)$. So, when the sets are equal to each other the implication is true, right? (Wondering)
 
  • #11
mathmari said:
Ah if $A=B$ we have that $A\subseteq B=A$ and that $f^{-1}(B)=f^{-1}(A)\subseteq f^{-1}(A)$. So, when the sets are equal to each other the implication is true, right?

Yep. (Nod)
 
  • #12
Thank you very much! (Mmm)
 

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