Advanced Discrete Mathematics: Sets (In dire need of help!)

[takamina]

Good afternoon!

I'm completely flabbergasted with some question I need to answer for my discrete mathematics course.
Any help would be greatly appreciated!
These are set related proofs, and are quite...advanced I suppose?

First one is a Proof: (aka prove or disprove

1. For all sets X ,Y, and Z:
   if(Z\(X∩Y))=(X∩Z)∪(Y∩Z), then Z⊆Y∪X.
   

Second one is also another proof:

1. (∀x∈R) (x∈N⇔((2x∈N)∧(3x∈N)))
   
   
   

Also I have a question? Can ''A⊆∅ '' ever be true? Since, isn't the empty set by definition empty? (AKA no subset A could be found within).

 

[Ishuda]

Good afternoon!

I'm completely flabbergasted with some question I need to answer for my discrete mathematics course.
Any help would be greatly appreciated!
These are set related proofs, and are quite...advanced I suppose?

First one is a Proof: (aka prove or disprove

2. For all sets X ,Y, and Z:
   if(Z\(X∩Y))=(X∩Z)∪(Y∩Z), then Z⊆Y∪X.
   

Second one is also another proof:

2. (∀x∈R) (x∈N⇔((2x∈N)∧(3x∈N)))
   
   
   

Also I have a question? Can ''A⊆∅ '' ever be true? Since, isn't the empty set by definition empty? (AKA no subset A could be found within).

What are your thoughts? What have you done so far? Please show us your work even if you feel that it is wrong so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting

The empty set is a strange animal. It is a member of every set, even itself. Oh, and every member of the empty set is the richest rodent in Denmark and at the same time the poorest rodent in Copenhagen.