Stuck in proving a tautology using algebraic propositions.

Question

I have to prove p v ¬(p ∧ q) is a tautology using the algebraic propositions.

Here is how I tried to do, but incomplete.

Q. p v ¬(p ∧ q)

(p v ¬p) ∧ (p v ¬q) using Distributive Law

T ∧ (p v ¬q) using Complement Law

?

Here I'm stuck.

Thanks for the help!

Answer 1

Your first step isn't quite right: \begin{align*} p \lor \neg (p \land q) &\equiv p \lor (\neg p \lor \neg q) &\text{by DeMorgan's Law} \\ &\equiv (p \lor \neg p) \lor \neg q &\text{by Associative Law} \\ &\equiv \top \lor \neg q &\text{by Inverse Law} \\ &\equiv \top &\text{by Domination Law} \end{align*}