Two identities on set theory

Question

I was reading extension of measure from the book 'An Introduction to Measure and Integration' by I.K.Rana. In a theorem of that book he proves how to get a complete measure space from an ordinary measure space (theorem 3.11.8, page-92). In the proof of that theorem he has used two identities of set theories. I am unfamiliar to those identities and seems very nontrivial to me. I cannot prove the the identities even after few attempts. The identities are: $$E\cup N=(E\setminus A)\triangle (A\cap(E\cup N))$$ & $$E\triangle N=(E\setminus A)\cup (A\cap(E\triangle N)),$$ where $N\subset A$. Although I see from Vein diagram that the identities are correct. But I want a formal proof. Please help me out.

Answer 1

You can see that $E\setminus A=(E\cup N)\setminus A$, so we can simplify the first equality as $$C=(C\setminus A)\triangle (C\cap A),$$ where $C=E\cup N$. But $(C\setminus A)$ and $(C\cap A)$ are disjoint, so the symmetric difference is just a union. Similarly, you can see that $E\setminus A= (E\triangle N)\setminus A$, and you can simplify the equality.

Note that we need $N\subseteq A$ to prove both equalities.

Answer 2

You can usually do these things with indicator functions. For $A$ a subset of the universe, define $$1_A(x)=\begin{cases}1,&x\in A\\0,&x\notin A\end{cases}$$

Then we have $$\begin{align} 1_{X\cup Y}&=1_X+1_Y-1_X\cdot1_Y\\ 1_{X\cap Y}&=1_X\cdot1_Y\\ 1_{X\setminus Y}&=1_X-1_X\cdot1_Y\\ 1_{X\triangle Y}&=1_X+1_Y-2\cdot1_X\cdot1_Y\\ \end{align} $$ Also, since we are given $N\subset A$, we know that $$ 1_N\cdot1_A= 1_N$$ This gives a rather mindless way to verify set identities.