How Can I Prove the Equality of Union and Intersection in This Homework?

[chocolatelover]

Homework Statement

Prove (A is a union of B)/(A is an intersection of B)=(A/B) is a union of (B/A)

Homework Equations

The Attempt at a Solution

Could someone first help me translate all of this into plain English. I don't really understand what I need to prove. Would I start off with the contrapositive? Is the contrapositive "If (A/B) is not the union of (B/A), then A is not the union of B/(A is not the intersection of B) and it is not equal to the antecedent"? Could someone please show me where to go from here?

Thank you very much

 

[GregA]

Homework Statement

Prove (A is a union of B)/(A is an intersection of B)=(A/B) is a union of (B/A)

Homework Equations

The Attempt at a Solution

Could someone first help me translate all of this into plain English. I don't really understand what I need to prove. Would I start off with the contrapositive? Is the contrapositive "If (A/B) is not the union of (B/A), then A is not the union of B/(A is not the intersection of B) and it is not equal to the antecedent"? Could someone please show me where to go from here?

Thank you very much

I assume the question was given as:
$$(A \cup B)/(A \cap B) = (B/A)\cup(A/B)?$$
If x is an element of the set on LHS then x is in A or x is in B but x is not in both A and B
what can you say about RHS? does it imply something about x that will help you get LHS?

 

[andytoh]

This is called the symmetric difference of two sets. It can be proven the the associative, distributive, and commutative laws holds with symmetric difference. Those are good exercises.

 

[chocolatelover]

Thank you very much

I assume the question was given as:

That's correct, except A and B are switched in the second part. (A/B) U (B/A)

Would the contrapositive also prove it?

I know how to use the associative property, but I'm sure how how use the others to prove this. I know that, say, A upside B upside C=(A upside B) upside U C=A upside U (B upside U C) I'm not sure how to do that or the others for this problem. Would it be (A U B)/(A upside U B)=A U B/A upside U B?

Could some please help me on this?

Thank you