# Thread: How would you prove...

1. ## How would you prove...

How would you prove that

(A/B) U B = A if and only if B C A ?

It's the iff that is giving me problems. Is there a technique for this type of proof?

2. ## Re: How would you prove...

Originally Posted by Cairo
How would you prove that
(A/B) U B = A if and only if B C A ?
There is no one way to do any set theory proof.
Here you might note that $\displaystyle \left( {A\backslash B} \right) \equiv \left( {A \cap B^c } \right)$

Starting with $\displaystyle B\subseteq A$ what would $\displaystyle \left( {A \cap B^c } \right)\cup A=~?$

3. ## Re: How would you prove...

Originally Posted by Cairo
How would you prove that

(A/B) U B = A if and only if B C A ?

It's the iff that is giving me problems. Is there a technique for this type of proof?
This statement is equivalent to the following statement :

$\displaystyle (p\land \lnot q) \lor q \Leftrightarrow p$ iff $\displaystyle p \lor q \Leftrightarrow p$

4. ## Re: How would you prove...

Originally Posted by princeps
This statement is equivalent to the following statement :

$\displaystyle (p\land \lnot q) \lor q \Leftrightarrow p$ iff $\displaystyle p \lor q \Leftrightarrow p$
Thanks, but this looks even worse to me!

5. ## Re: How would you prove...

Would this not be A, Plato?

I was thinking of relabelling (B C A) as D and ((A/B) U B = A) as E and then trying to argue D implies E and also that ~D implies ~E, but not sure if this would work or even where to start.

6. ## Re: How would you prove...

Originally Posted by Cairo
Would this not be A, Plato?
That is correct. So you have proved it one way.

Now suppose that $\displaystyle (A\cap B^c)\cup B=A$ now show that $\displaystyle B\subseteq A.$

7. ## Re: How would you prove...

Originally Posted by Plato
That is correct. So you have proved it one way.

Now suppose that $\displaystyle (A\cap B^c)\cup B=A$ now show that $\displaystyle B\subseteq A.$
Using the distributive law I get

(A U B) and (B^c U B) = .......

But not sure how to continue this argument.

Will have a think about it.

8. ## Re: How would you prove...

Originally Posted by Plato
There is no one way to do any set theory proof.
Here you might note that $\displaystyle \left( {A\backslash B} \right) \equiv \left( {A \cap B^c } \right)$

Starting with $\displaystyle B\subseteq A$ what would $\displaystyle \left( {A \cap B^c } \right)\cup A=~?$
Should not $\displaystyle \left( {A \cap B^c } \right)\cup A$ be $\displaystyle \left( {A \cap B^c } \right)\cup B$

And then be equal to: $\displaystyle (B\cup A)\cap(B\cup B^c)$ ?