# Is the following statement equivalent?

Is the following statement equivalent: $((A \vee \neg C) \wedge ((\neg B) \leftarrow C)) \vee \neg (A \wedge B) \equiv (\neg A \vee \neg B \vee \neg C)$

Our prof gave us an exercise with this statement and said we should simplify it with truth table. Could that be the right solution? Hope somebody can help.

• Did you try the truth table approach? With three variables, it'll be fairly simple. – Rick Decker Feb 13 '16 at 19:57

We have $¬B ← C ≡ ¬B ∨ ¬C$, so $(A ∨ ¬C) ∧ ((¬B) ← C)) ≡ (A ∨ ¬C) ∧ (¬B ∨ ¬C) ≡ (A ∧ ¬B) ∨ ¬C$. We have also $¬(A ∧ B) ≡ ¬A ∨ ¬B$, and hence the original proposition is equivalent to $¬A ∨ ¬B ∨ ¬C ∨ (A ∧ ¬B)$, which is just $¬A ∨ ¬B ∨ ¬C$ since $A ∧ ¬B$ implies $¬B$.