prove that for sets

if and only if

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- Dec 1st 2009, 07:20 PMsantiagos11symmetric difference of sets
prove that for sets

if and only if - Dec 1st 2009, 09:11 PMDrexel28
- Dec 2nd 2009, 04:47 AMemakarov
There is also a different way to solve this that sometimes works very well with symmetric difference.

Define the function by if and if . Here is any set and is any object.

Also, define the function as follows:

(If we associate 1 with True and 0 with False, then . Also, is addition modulo 2.) One can check that is commutative and associative.

Show that for all , , and .

The part above was generic; it applies to any problem with symmetric difference and has to be done only once.

Now, from this problem's assumption and the fact above we have for all . We can add (using ) an expression to both sides where is , , , or . E.g., if we add to both sides, we get = . From here it is easy to show the desired equation. - Dec 2nd 2009, 05:05 AMGrandad
Hello santiagos11I think this is easiest to prove by re-writing it as the equivalent statement using logical propositions. In other words:

Prove that if are logical propositions, then if and only if , where denotes 'exclusive or'.This is quite straightforward if you use a Truth Table for each of the compound propositions and show that each table produces an identical output.

The attached diagram shows the truth table for . It is a simple matter to show that the table for produces the same result.

Grandad