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Prove of disprove: for every 3 sets, $A,B,C: (A\setminus B)\cup (A\setminus C)=B\Longleftrightarrow A=B$ and $B \cap C = \emptyset$

Proof : We prove each statement implies the other.

1) First we show that $(A\setminus B)\cup (A\setminus C)=B\Longrightarrow A=B$ and $B \cap C = \emptyset$

2) $(A \setminus B)=A$ and $(A \setminus C)=A$, therefore $A=B$

3) Since $A=B$, then $(A\setminus C) = (B\setminus C)$

4) Therefore, $B\cap C = \emptyset$

5) Now we show that $A=B$ and $B\cap C = \emptyset \Longrightarrow (A\setminus B)\cup (A\setminus C)=B$

6) Since $A=B$, then $(A\setminus B) = \emptyset$

7) $B\cap C$ must not share any common elements for $B\cap C = \emptyset$ to hold

8) Thus, $(A \setminus C) = (B \setminus C) = B$ (since $A=B$)

9) Therefore $(A \setminus B) = \emptyset$ and $(A \setminus C) = B$

10) This gives us that: $\emptyset \cup B = B$

I would to like know if my proof is complete, correct and where it contains any faulty logic. I'm new to proof writing, so constructive criticism goes a long way for me.

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  • $\begingroup$ How is 2) related to 1)? Note that $X\setminus Y = \{s\,|\,s\in X,s\notin Y\}$. $\endgroup$ – Nex Nov 10 '15 at 19:16
  • $\begingroup$ @Nex The set differences. No? $(A \setminus B)\Longrightarrow A$ and $(A \setminus C)\Longrightarrow A$ $\endgroup$ – Cherry_Developer Nov 10 '15 at 19:18
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We have: $x \in B \Leftrightarrow x\in A\setminus C$ (because $x \notin A\setminus B$) $\Rightarrow A\setminus C=B \Rightarrow B=(A\setminus B)\cup(A\setminus C)=A\setminus B \cup B = A$.

On the other hand, $A\setminus C = B \Leftrightarrow B\setminus C = B \Leftrightarrow C= \emptyset \Rightarrow B\cap C=\emptyset$

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  • $\begingroup$ Can you please clarify this? $\endgroup$ – Cherry_Developer Nov 10 '15 at 20:05
  • $\begingroup$ can you specify which part? $\endgroup$ – SiXUlm Nov 10 '15 at 20:37
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Why does $(A \setminus B) = A$ in step 2) ? There is no explanation (it's false too, in general). $(A \setminus C) = A$ is similarly cryptic (and false), and why does it follow from these that $A = B$ ??

Why does it follow from $A \setminus C = B \setminus C$ that $B \cap C = \emptyset$ ? One can easily come up with counterexamples for this implication.

To see one implication: suppose $(A \setminus B) \cup (A \setminus C)=B$.

We want to show $A = B$. Suppose $x \in A$. Then, if $x \notin B$ were true, $x$ would be in $A \setminus B$, so in $(A\setminus B)\cup (A\setminus C)=B$ and so $x \in B$, and this is a contradiction ($x \notin B$ implies $X \in B$). So $x \in B$ must hold and this shows that $A \subseteq B$. On the other hand, suppose $x \in B = (A\setminus B)\cup (A\setminus C)$. Then $x \notin (A \setminus B)$, so $x \in A \setminus C$, so in particular $x \in A$. This shows that also $B \subseteq A$ and hence $A = B$ from the two inclusions.

We also want to show $B \cap C = \emptyset$. If $x \in B \cap C$, then $x \in B = (A\setminus B)\cup (A\setminus C)$. As $x \in C$, we know that $x \notin A \setminus C$. So it must be the case that $x \in A \setminus B$, but this cannot be as $x \in B$. So such an $x$ cannot exist, and the intersection is empty.

The reverse implciation is similar.

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  • $\begingroup$ So the entire statement is false? $\endgroup$ – Cherry_Developer Nov 10 '15 at 20:16
  • $\begingroup$ $A \setminus B = A$ is, yes. The statement to prove is true, I think (the bi-implication), I showed one half. The steps themselves are not. You have to start thinking interms of proving inclusions, etc. See my proof of one half, which I wrote out as using the most elementary notions. $\endgroup$ – Henno Brandsma Nov 10 '15 at 20:20
  • $\begingroup$ Yes, what you wrote out is very helpful. Can I write up the entire proof and then have you look it over? $\endgroup$ – Cherry_Developer Nov 10 '15 at 21:01
  • $\begingroup$ Would you please be able to clarify where you showed that $A = B$. I'm confused by what you meant exactly. I'm especially confused by: so x∈B, and this is a contradiction (x∉B implies X∈B). $\endgroup$ – Cherry_Developer Nov 10 '15 at 23:28

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