How do you go from $(\lnot b \land \lnot c) \lor (b \land c)$ to $\lnot(b \oplus c)$ elegantly?

Question

I have: $(\lnot b \land \lnot c) \lor (b \land c)$

I want to translate this into $\lnot(b \oplus c)$ but I can't seem to get there elegantly.

My attempt:

$(\lnot b \land \lnot c) \lor (b \land c)$

$\lnot(\lnot(\lnot b \land \lnot c) \land \lnot(b \land c))$

$\lnot((b \lor c) \land (\lnot b \lor \lnot c))$

$\lnot((b \land (\lnot b \lor \lnot c)) \lor (c \land (\lnot b \lor \lnot c)))$

$\lnot((b \land \lnot b) \lor (b \land \lnot c) \lor (c \land \lnot b) \lor (c \land \lnot c))$

$\lnot(\text{False} \lor (b \land \lnot c) \lor (c \land \lnot b) \lor \text{False})$

$\lnot((b \land \lnot c) \lor (c \land \lnot b))$

$\lnot(b \oplus c)$

I think this is correct but it felt so longwinded and I suspect I made an error somewhere anyway, I don't know why. It feels "inelegant".

Is there a more direct way to get there?

Answer 1

The shortest way is$$[(\lnot b \land \lnot c) \lor (b \land c)]=[b=c]=[\lnot(b \oplus c)].$$

Answer 2

If the only equivalence regarding $\oplus$ that you have is:

$$b \oplus c \Leftrightarrow(b \land \neg c) \lor (c \land \neg b)$$ then what you do is pretty much the only thing to do.

But some texts have as an equivalence:

$$b \oplus c \Leftrightarrow (b \lor c) \land \neg (b \land c)$$

and with that, it would just be a simple DeMorgan:

$$(\lnot b \land \lnot c) \lor (b \land c) \Leftrightarrow$$

$$\neg (b \lor c) \lor \neg \neg (b \land c)) \Leftrightarrow$$

$$\neg ((b \lor c) \land \neg (b \land c)) \Leftrightarrow$$

$$\neg (b \oplus c) $$

Answer 3

Note that $x\oplus y$ means $x$ or $y$ but not both. In logical symbols $x\oplus y= (x \lor y) \land \neg(x \land y)$

$\neg(x\oplus y)=\neg[ (x \lor y) \land \neg(x \land y)]$

Applying De Morgan we have

$\neg(x\oplus y)= \neg(x\lor y) \lor (x\land y)$.

And we are at the desired result after one more application of De Morgan.

Maybe the best piece of advice is to start with the Right Hand Side rather than the left. It's a little easier to work with.