Showing logical equivalence of these two formulas

Question

I have the following statement in propositional logic:

(¬g v s1 v ¬s2) ^ (¬g v ¬s1 v s2) ^ (¬g v s1 v s2) (1)

I want to show equivalence to this statement:

(¬g v s1) ^ (¬g v s2) (2)

I can use the distributive property on (1) to obtain the following statement:

(¬g v s1 v ¬s2) ^ (¬g v ¬s1 v s2) ^ ((¬g v s1) v (¬g v s2)) (3)

I can see that the last conjunct renders the first two 'redundant', but I do not know that steps to take to show logical equivalence to (2).

Can anyone help?

Thank you

Answer 1

The formula :

(2) $(¬g \lor s_1) \land (¬g \lor s_2) $

is equivalent to [recall that : $p \lor F \equiv p$ and $p \land \lnot p \equiv F$]:

$[(¬g \lor s_1) \lor (s_2 \land ¬s_2)] \land [(¬g \lor s_2) \lor (s_1 \land ¬s_1)]$.

By Distributive property we have :

$(¬g \lor s_1 \lor s_2) \land (¬g \lor s_1 \lor ¬s_2) \land (¬g \lor s_1 \lor s_2) \land (¬g \lor ¬s_1 \lor s_2)$.

Removing the repeated term we have :

(1) $(¬g \lor s_1 \lor s_2) \land (¬g \lor s_1 \lor ¬s_2) \land (¬g \lor ¬s_1 \lor s_2).$