Solving Set Equality Proof Homework

[Haydo]

Homework Statement

Let $A, B, C$ be sets with $A \subseteq B$. Show $(A-B)\cup C=(A\cup C)-(B\cup C)$

Homework Equations

None.

The Attempt at a Solution

So, generally, one shows two sets to be equal by showing that each is a proper subset of the other. I started with the LHS. Thus, if x is in (A-B)UC, x is in (A-B) or x is in C. But if x is in C, then x is not in RHS. So it seems that the expression does not hold. Am I thinking of this wrong, or did the assignment writer make an error?

 

[Geofleur]

If x is in C, then x can still be in the right hand side. It might help to draw a Venn diagram.

 

[Haydo]

I should say, consider the case where x is in C and x is not in (A-B). Then x cannot be in (AUC)-(BUC), right?

 

[Geofleur]

You know what, I think you're right! If x is in C, then for x to be in the RHS it could not be in $B \cup C$. But then it cannot be in $C$. Also, my first Venn diagram had a mistake in it.

 

[Haydo]

Hm. That's what I thought. I'll email the professor then. Maybe he was just trying to see if we were paying attention, but the problem definitely said to prove the equality, not to prove it or show it is false.