Boolean Algebra - is this statement correct?

Question

So I have a statement that goes like this:

$$ ( \lnot A \lor B) \land(\lnot A \lor \lnot B) $$

I think it is equivalent to

$$ \lnot A $$

Am I right or not?

Answer 1

Although it's a bit overkill for this problem, since there are only two variables, you can easily solve this with a truth table:

A	B	$\lnot A$	$( \lnot A \lor B) \land(\lnot A \lor \lnot B)$
True	True	False	False
True	False	False	False
False	True	True	True
False	False	True	True

As you fill in the last column, you can quickly realize that the truth value of B has no impact on the truth value of the expression $( \lnot A \lor B) \land(\lnot A \lor \lnot B)$.

Answer 2

1. We have $(¬A∨B)∧(¬A∨¬B)$

2. Realize that, by using the distributive law that is $p∨(q∧r)≡(p∨q)∧(p∨r)$, $-A∨(B∧¬B)$ is logically equivalent to$(¬A∨B)∧(¬A∨¬B)$. Let's have the simpler equivalent: $¬A∧(B∨¬B)$

3. Using negation law yields $¬A∧⊤$

4. Using identity law yields $¬A$

So, yes, you're right: $(¬A∨B)∧(¬A∨¬B)≡¬A$