Why is this clause unsatisfiable?

Question

The question says, Which of the following statements apply?

And the answer says this one does but I'm not sure why.

• The clauses {a,b},{a,-b},{-a} are unsatisfiable
• The clauses {a},{-a,b},{-b} are unsatisfiable

Also, why is it not possible to derive the empty clause from {x,y},{-x,-y}?

Answer 1

Why is it not possible to derive the empty clause from $\{ x, y \}, \{ \lnot x, \lnot y \}$ ?

Because $\{ x, y \}$ is a clause, i.e. a disjunction : $x \lor y$.

Thus the couple of disjunctions :

$x \lor y \ $ and $ \ \lnot x \lor \lnot y$

is obviously satisfiable; try with a truth assignment $v$ such that :

$v(x)=$t and $v(y)=$f.

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If we apply the Resolution rule to : $\{ x, y \}, \{ \lnot x, \lnot y \}$, starting, e.g. with $x$ and $\lnot x$ (but the choice is immaterial), what we get is the clause :

$\{ y, \lnot y \}$

and this is not the empty clause. If we "read" it as : $y \lor \lnot y$, it is obvious that it is satisfiable.