Is this statement true or false?

[Frankenstein143]

Hello everyone,
unfortunately I do not understand how to solve this problem:

Let A and B be subsets of a set X. Prove generally the correctness of statements (a) to (c) or refute the corresponding statement by giving a concrete counterexample.
(A \ B) ∪ (B \ A) = (A ∪ B) \ (A ∩ B)

I tried to solve this problem but then came to the conclusion that the answer depends on the Assumption whether A intersects B.
If A intersects B then it is true. If A do not intersects B then it is false.
I I right or wrong? please help

 

[Dr.Peterson]

Hello everyone,
unfortunately I do not understand how to solve this problem:

Let A and B be subsets of a set X. Prove generally the correctness of statements (a) to (c) or refute the corresponding statement by giving a concrete counterexample.
(A \ B) ∪ (B \ A) = (A ∪ B) \ (A ∩ B)

I tried to solve this problem but then came to the conclusion that the answer depends on the Assumption whether A intersects B.
If A intersects B then it is true. If A do not intersects B then it is false.
I I right or wrong? please help

Can you give an example where it is false? That's what they're asking for.

And can you represent the question with a Venn diagram (which would include the intersection)?

 

[pka]

Let A and B be subsets of a set X. Prove generally the correctness of statements (a) to (c) or refute the corresponding statement by giving a concrete counterexample.
(A \ B) ∪ (B \ A) = (A ∪ B) \ (A ∩ B)

Do you know about the symmetric difference of two sets? And here

 

[lex]

You could prove it by:

[MATH] \hspace2ex (A \cup B)\setminus(A\cap B)\\ =(A \cup B) \cap \overline{(A \cap B)}\\ =\boldsymbol{(A \cup B)} \cap (\overline{A} \cup \overline{B})\\ =(\boldsymbol{(A \cup B)} \cap \overline{A}) \cup (\boldsymbol{(A \cup B)} \cap \overline{B}) ... [/MATH]
Or you could prove that:
[MATH](A \setminus B) \cup (B \setminus A) = (A \cup B) \setminus (A \cap B)[/MATH]by taking [MATH]x \in (A \setminus B) \cup (B \setminus A)[/MATH] and proving that [MATH]x \in (A \cup B) \setminus (A \cap B)[/MATH]and then taking [MATH]x \in (A \cup B) \setminus (A \cap B)[/MATH] and proving that [MATH]x \in (A \setminus B) \cup (B \setminus A)[/MATH]