Verifying simplification of simple logic expressions

Question

I am trying to simplifying these 3 expressions, and I have actually done them, but can someone check if it's correct? Thank you.

• $x \vee y \land y \vee \neg x$
• $x \land y \vee y \land \neg x$
• $\neg(\neg(x \land y) \vee x)\vee y$

The first one:

$x \vee y \land y \vee \neg x = x\vee(y)\vee\neg x = (x\vee\neg x)\vee y=1\vee y=1$

Second one:

$x\land y \vee y \land \neg x = (x \land y) \vee (y \land \neg x) = (y \land x) \vee (y \land \neg x) = y \land (x \vee \neg x) = y \land 1 = y$

Third one:

$\neg(\neg(x\land y)\vee x)\vee y = \neg(\neg x \vee \neg y \vee x) \vee y = \neg(1\vee\neg y)\vee y=\neg(1)\vee y=0\vee y=y$

Thank you for helping.

Answer 1

Yes, those are all correct.   Also, your steps are clear, so it was easy to follow your reasoning.