Question

prove
that
(A-B)-C = (A -c) - B = A-(BUC)​

Answer 1

Answer with Step-by-step explanation:

We have to prove that

(A-B)-C=(A-C)-B=A-(BUC)

Suppose $x\in (A-B)-C$

Then $x\in(A-B)$ and $x\notin C$

Then, $x\in A$ but $x\notin B$

Therefore, $x\in(A-C)$ and $x\notin (B\cup C)$

Hence, $x\in(A-c)-B$ and $x\in A-(B\cup C)$

Conversely ,

Let $x\in (A-C)-B$

Then, $x\in(A-C)$ but $x\notin B$

Therefore, $x\in A$ and $x\notin C$

Therefore, $x\in (A-B)-C$

And $x\notin (B\cup C)$

Therefore, $x\in (A-(B\cup C))$

If$x\in A-(B\cup C)$

Then, $x\in A$ but $x\notin (B\cup C)$

Therefore, $x\notin B$ and $x\notin C$

Then, $x\in (A-B)-C$ and $x\in (A-C)-B$

Hence, $(A-B)-C=(A-C)-B=A-(B\cup C)$