welcome back to logic 101 I'm William Spaniel and this lecture is on the exclusive or which you will recall means an or expression where one part is true and the other part is false and in my introduction to the inclusive or I said that there's a way to write down the exclusive or using just The Logical notation that we already have so what I want to do in this lecture is twofold first I want to show you how to write down the exclusive or using just The Logical notation that we already have and second I want to verify that that is the way to write it down and I'm going to verify it by using a truth table so we get a little bit of extra practice in here with those truth tables let's get to it this is how you write down the exclusive or you say P or q and not p and Q why is that the exclusive or well it might not be so easy to see by just staring at it so let's work on a truth table and verify that that is correct so I've done a little bit of the work already here we have two simple sentences p and Q I've gone ahead and filled in all the combinations of truth values for those two two simple sentences and then in the last column we have the overall expression P or q and not p and Q so if that is in fact the exclusive or what that should recover is a truth value of false in the top row and the bottom row and a truth value of true in the middle two rows so it should be false in the top row because both of those things are true and it should be false in the bottom row because both of them are false whereas it should be true in the middle two rows because one is true and the other one is false in those middle two rows so that's what we should expect out of our truth table if in fact that last column does represent an exclusive or so let's go through this process of creating a truth table to actually get to that so the other three columns are filled in like that so we have on the left side of the conjunction uh disjunction between p and Q so that's what we see in column 3 and on the right side of the overall conjunction we have the negation of p and Q so we start off with the non-negated version so we have just regular p and Q there and then we in the next column over negate that fourth column so we have not p and Q in the fifth column and then that unlocks the overall expression which is the conjunction of column 3 and column 5 so now that we have those things filled in we can go through each column and fill in the truth values for those columns so let's start with column three here we have a disjunction between p and Q remember that a disjunction is true when at least one part is true it's false when both parts are false so that means we have a truth value of true in the first three rows and a truth value of false in just the bottom row for the fourth column we have a conjunction remember remember that a conjunction is true only when both of the component parts are true it's false when at least one of them is false so that means we have a truth value of true in the top row and a truth value of false in each of the bottom three rows for the fifth column we have a negation of the fourth column so that is essentially flipping the on andof switch in the fifth column from the fourth column so we take the fourth column and we take all values of true and switch them over to false and we take all values of false and switch them over to being true and if you do that then we just have that there so we have a false value in the top row and a true value in the bottom three rows and that leaves us with the overall expression which I told you is an exclusive or so it's p or q and not p and Q and that's a conjunction of columns three and five so for any other conjunction just like any other conjunction we look to see when both of those columns are true and we mark down those as being true and when at least one of them is false we mark it down as being false so if we look at columns 3 and five we see that columns three and five are both true only in the middle two rows so we give True Values to the middle two rows we give false values to the top row and the bottom row that's because in the first row column five is false and in the bottom row column three is false so in those two cases in the top case and the bottom case we have have at least one part of the conjunction being false which means the conjunction is false whereas in the middle two cases we have both component parts being true so the conjunction is true and that is why that expression P or q and not p and Q is in fact an exclusive or so we got a little bit more practice in with truth tables and that's good and we have a little bit ways to go on Truth tables in this unit so join me next time when we get to one of those last parts of Truth tables take care