What is the difference between proving an equation versus verifying LHS = RHS?

Question

I am referring to the book Mathematical Proofs: A Transition to Advanced Mathematics (3rd Edition) . In the chapter 4.4, I came across the following example -

Prove that for every two sets A and B, (A ∪ B) − (A ∩ B) = (A − B) ∪ (B − A).

The solution said that to prove the above statement is true, we need to show that LHS ⊆ RHS and RHS ⊆ LHS.

My question is that, can I answer the question by simplify both, LHS and RHS, to show LHS = RHS? Will that be enough to answer the question?

Thanks!

Answer 1

You're getting at two different ways of proving that two sets are equal.

One way to show two sets $X$ and $Y$ are equal is to show that $X \subseteq Y$ and $Y \subseteq X$.

Another way is to apply the laws of set algebra (e.g. De Morgan's law) and definitions (e.g. definition of set difference). In this way, you're "simplifying" both sides.

In general, both of these are valid ways of proving two sets are equal. In this case, I would suggest trying both ways. The first way will give you more intuition in understanding the definition of set differences, unions and intersections. The second way will help you practice utilising the laws of set algebra (in particular, you may have to utilise distributivity so your expression will get a bit complicated).