# Propositional Logic, Resolution - derived tautology from 2 clauses of cardinality 3...

I have 2 clauses of cardinality 3

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

or, in set notation

$$\{a, b, c\}, \{a, \lnot b, \lnot c\}$$

I applied resolution incorrectly and got clause

$$\{a\}$$

Now, someone told me that if we apply resolution on these 2 clauses we get a tautology, but I don't understand why/how.
Can someone please explain? Thank you.

You only resolve on one literal. This was your mistake: you resolved on both $$b$$ and $$c$$, but you can only resolve on one of them.
Now, if you resolve on $$b$$, the result is $$\{a,c,\neg c\}$$, which is indeed a tautology, since whether $$c$$ is true or false, this clause will always be satisfied: it is a tautologous clause.
Likewise, if you resolve on $$c$$, the result is $$\{a , b , \neg b\}$$, which is a tautologous clause as well.
• I'm confused on how you get a tautology here. If $a$ is false and either $b$ and $c$ are true or $b$ and $c$ are both false, won't the statement be false: $(F \vee T \vee T) \wedge (F \vee F \vee F)$? It would seem to me that this simplifies to $a \vee (b \oplus c)$. Feb 9 '19 at 2:53