Showing SAT is in NP: Given a truth assignment to all the propositions, we need to check one by

one if each of the conjunct is true or not. This can be done in time linear (thus polynomial) in the

size of the input formula. Hence, Satisfiability is in NP.

To show SAT is NP-complete we now need to show that every problem in NP can be expressed

as the satisfiability of a propositional formula. Since any problem in NP can be solved using a

non-deterministic Turing machine, all we need is to express the operation of a Turing machine as a

formula. Suppose the non-deterministic Turing machine has δ(q, X) = {(q

0

, Y, L),(q

00, Y 00, R)}. This

can be expressed by a formula A ⇒ B ∨ C where the proposition A means that the current state is

q and the tape symbol is X. B means that the current state is q

0

, the tape symbol to one step right

is Y

0

. C means that the current state is q

00, the tape symbol to the left is Y

00. (This is just a sketch

of the proof.)