Sets

[Max.Maxim]

I don't know if I have done it correctly or not.

 

[tkhunny]

How did we get [math]A = A\cap B[/math]?

 

[Max.Maxim]

I think I meant to say B=A∩B.

 

[Steven G]

You had a few things that I did not like.

in the end you NEVER SAID THAT ALL THIS MEANS ... Major error!

You wrote so when you meant to write show AND you wrote shows when you meant to write show

 

[HallsofIvy]

After showing that \(\displaystyle B\subseteq A\) you say "In converse let that \(\displaystyle A\subseteq B\). "let" implies that this is part of your hypothesis. It isn't! It is the conclusion that you want to prove. The hypothesis for the converse is that \(\displaystyle B\subseteq A\).

But you may have confused the "converse" with simply proving both parts of "\(\displaystyle A\cap B= B\)". Since "X= Y" implies both "\(\displaystyle X\subseteq Y\)" and "\(\displaystyle Y\subset X\)", you really need to prove four things:
1) if \(\displaystyle A\capeq B= B\) then \(\displaystyle A\subseteq B\).

2) if \(\displaystyle A\capeq B= B\) then \(\displaystyle B\subseteq A\).

3) if \(\displaystyle A= B\) then \(\displaystyle A\cap B\subseteq B\).

4) if \(\displaystyle A= B\) then \(\displaystyle B\subseteq A\cap B\).