Symmetric difference problem (Real Analysis)

[phillyolly]

Homework Statement

What am I asked to do in the problem? Am I just asked to draw a diagram or to prove a) and b)?

Homework Equations

The Attempt at a Solution

 

[Raskolnikov]

Draw a Venn diagram with sets A and B. Shade in the area "that belongs to either A or B, but not both."

Then, I'd say just make sure you first know exactly what (a) means, i.e., what is A\B and B\A. You should easily be able to see that the definition of D in (a) is the same as the more intuitive definition "either A or B, but not both."

Finally, you should write a quick formal proof that shows the definition in (a) is the same as that in (b).
i.e, prove that (A \ B) U (B \ A) = (A U B) \ (A $$\cap$$ B).

Since you're new to all of this, don't skip steps. Take care of the details!

 

[phillyolly]

Thank you...

Based on your suggestions...I tried...Where is my mistake?

 

[Raskolnikov]

hehe, first of all, you have A/B instead of A\B.
Secondly, your second line is utterly meaningless. The union operation "U" is only defined on sets, not on boolean statements such as "x is element of A\B."
Thirdly, $$x \in A \backslash x \in B$$ is also utterly meaningless. The relative complement operation "\" is defined only on sets.

Since you are new to all this, break this down to the very basics.
$$x \in (A \backslash B) \cup (B \backslash A).$$
$$x \in (A \backslash B) \vee x \in (B \backslash A).$$
$$(x \in A \wedge x \notin B) \vee (x \in B \wedge x \notin A).$$

Try completing from there. Step-by-step! I know a lot of it seems like just grunt work, but it's a very important fundamental approach :P!