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

Question

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.

Answer 1

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.