Transforming logical expressions, De Morgan's laws. De Morgan's laws tell us how to transform in a expression like naught P and Q which is equivalent to naught P or naught Q. We give to prove. It's very simple, just another truth table, naught P and Q. Well, P can be either true or false. Q can be either true or false. If P and Q are both true, then P and Q is true as well. Naught P and Q would then be false, so I've got a 0 here. If P is false, then P and Q is force as well. And naught P and Q is true, so I've got a 1 here. And the same holds bisymmetry for Q. And if P and Q are both false, then P and Q is false, and not P and Q is true. So this is the truth table of naught P and Q. And the truth table of the expression on the left hand side, naught P or naught Q. We know that an or expression is true if either the left hand side or the right hand side is true, well, the left hand side is true. If P is false, the right in side is true if Q's false. And the only possibility of the expression being false is that those P and Q are true, and we see those are the same truth table. So this is the way we prove the first law of De Morgan. So let's have a quick example. If we have the sentence, it is not true that there is a tiger in the left room, and a tiger in the right room. Then this is equivalent to saying, either there is no tiger, In the left room, Or no tiger in the right room. Of course, it would be good to know which is which. There's of course a second variant of De Morgan's law for the case of or. If it is not true that P or Q is true, that is equivalent of saying P is not true, and Q is not true. Now, let's look at a quick question. If you have the expression, P, and naught P, and Q, can you transform this into a simple expression? So, let's try it. P, and naught P, and Q, we used De Morgan's law to work out the right end terms, naught P or naught Q. Then we take our distributive law, and say, well, this is P and naught P, or P and naught Q, but P and naught P that can never be true, that's always false, or P and naught Q. Well, false or statement has the same truth value as the statement itself. So we end P and naught Q have to be both true. Now we can prove a logical, Law, because we've got all the necessary tools in place. And the logical law, you can now prove is the following. If you know that P is true, and that P implies Q, then this will imply Q. Well, this will better be true, otherwise, logic doesn't have lot sense for us because it wouldn't respond to the every day usage of the words. But now, let's see whether we can transform this law in such a way that we actually can say something about the truth value. The first thing we do is to get rid of the implication sign on the right. We know that a statement implies another statement. Then either this statement is false or that statement is true. We use this by writing naught P, and P implies Q or Q. All right, the next item of business is to get rid of the second implication. So this is naught P, and, and again, this we can write as naught P, or Q. So, this is already a lot cleaner. It's now a good idea to work this out using the distributive law. So we have naught P and naught P, or P and Q, and we still have our Q over here. Well, we are rewarded for doing this because P and naught P is always false. So this is false statement, or a null statement. So the only thing which remains of this is the second statement. So we have naught P and Q, or Q. And now we can use De Morgan's law, and that will give us naught P, or Q, Or Q. Correction, naught P, or naught Q, or Q. And since, by associativity, these brackets really don't matter. We've got naught P, or naught Q, or Q. But naught Q or Q is always true. So we have here really naught P or true, and this is finally equivalent to true. So we see that our original logical law is always true irrespective of what P and Q is.