 Sometimes we need to consider the negation of a conjunction. We have cookies and cream. We do not have cookies and cream. Or we might need to consider the negation of a disjunction. You can have ice cream or cake. You cannot have ice cream or cake. Let's consider what these statements might be logically equivalent to. To begin with, let's see if we can find a statement logically equivalent to the negation of the disjunction a or b. And so let's construct our truth table. And let's think about this a little. We see that most of the time the negation of a or b is false. And this occurs a lot of times with conjunctions. In most cases conjunctions will be false. Now in order for our conjunction to be true, both statements have to be true. And so here we see that if this is going to correspond to some sort of conjunction, the only time it's true is here. And in order for both of the statements to be true, the two statements have to be not a and not b. So if we negate a and b, we see that not a and not b will be true when a and b are false. And so the negation of a disjunction looks to be the conjunction of the negations. Let's verify that by filling out the rest of the truth table. And so we see that the negation of the disjunction does have the same truth values as the conjunction of the negations. And this leads to a very important equivalence that's usually known as de Morgan's law or at least de Morgan's law part one. The negation of a disjunction is the conjunction of the negations. Not a or b is the same as not a and not b. So for example, we do not have cookies or cream. Well, we can rewrite it. This is the negation of the statement, we have cookies or we have cream. This is actually the disjunction a or b. And de Morgan says that when we negate a disjunction, we get a conjunction of the negations. And so this will become not a and not b. We do not have cookies and we do not have cream. How about the negation of a conjunction? Well, let's construct our truth table. So the negation of a conjunction appears to be true in most cases. Since it's mostly true, it looks like a disjunction. Now remember, the only time a disjunction is false is when both of its component statements are false, which suggests we might want to look at not a and not b and the disjunction not a or not b. And we'll complete the rest of our truth table to verify their equivalence. And so it appears that the negation of the conjunction is the same as the disjunction of the negations. If one of them is false, so is the other. And if one of them is true, so is the other. And so we would say they are logically equivalent. And this gives us the second of De Morgan's rules. The negation of a conjunction is the disjunction of the negations. So if we may rewrite the statement, you may not have cake and ice cream. And this appears to be the negation of you may have cake and you may have ice cream. And that is a conjunction. You may have cake and you may have ice cream. And so De Morgan says that the negation of a conjunction is the disjunction of the negations. And so this will become not a or not b. You may not have cake or you may not have ice cream.