first replacement 4 mp4

Hi everyone, we're up to our fourth replacement rule this one's called distribution. And this one is one that's a little bit more tricky for people, so we're spending some extra time on it. Again, two sets of expressions that mean the same thing. In the first one, we have, "Gil can have ice cream AND he can have either cake or pie." That means the same thing as he can have ice cream and cake OR he can have ice cream and pie. Okay, so in shapes what happens with this one is we move from a statement whose main operator is a dot that's a dot between a single element and on the right side something that is a wedge. So we move from box dot circle or a triangle and we move to the main operator was the dot the main operator is gonna become the wedge. And that becomes box dot circle, so notice here that we're gonna pair up the box and the circle the main operator switches from a dot to a wedge and then we're gonna pair up the box and the triangle. Box dot triangle. So one way to think about this is, we're distributing this box to the circle and to the triangle. So we get box circle together, box triangle together. And the main operator changes from the dot to the wedge and the next main operator, what was the wedge, that becomes the main. The main becomes the smaller one. And a similar thing happens with our next set. So this is the first set. And with the next set we have, "Gil can either eat his dinner OR he can miss desert and go to bed." And that means the same thing as, "Gil can either eat his dinner or miss desert AND he can either eat his dinner OR go to bed." So notice a similar thing is happening. We're going to distribute that "D," that first element to the "M," "D" and "M" are together and then the "D" is going to distribute to the "B" as well. The "D" and the "B" are together. The wedge was the main operator here It becomes the smaller operator, it goes inside the parentheses. And what was inside the parentheses becomes the main operator. A similar thing is happening in both distributions. We distribute the first to the two and the main operator becomes subordinate, the subordinate becomes the main. I'll do this again in shapes: box, wedge, circle and triangle Means the same thing as box wedge circle and box, wedge, triangle. Remember too that this on, just as all of our replacements works in both directions. So, in a proof when you see something like either of these, where we have a dot between two wedges and that first element is the same you can pull that out and get the other two together. The shorthand for distribution is "DIST." Distribution. And let's look at some examples. And let's try and keep those on here for now. Alright, so here in the first one I have a dot and a wedge, so this is looking like this one here and did I transform it in just the same way? The "A" and the "B" go together, the "A" and the "D" go together. The dot should become the subordinate and look: it stayed the same. So this is not distribution. In order for it to be distribution, this dot needed to be a wedge and these wedges needed to be dots. Let's look at the second one. So here we have a wedge between two conjuncts so that's looking like this up here and notice those first conjuncts are the same, so we have box. So did I pull that one out? I did. I got a dot. That's what I want. And then the other two are now together. B wedge D, so that's exactly what I did, so this one here is making this move from here to here. So we do have a distribution on that one. Let's look at our final example. And what do we have here? Much more complicated. But this will be our box. This will be our circle. And this will be our triangle. So the A dot B together is what's in the box. The "D" is the circle. The "C" is the triangle. So I am moving from box, wedge circle and triangle and I move from there to box, OR circle AND box OR triangle and is that the distribution move? I have a wedge and a dot. The box and the circle go together Box and circle together. The box and triangle together. Box and Triangle together. What was the main operator becomes the smaller one. What was the smaller one becomes the main operator. So yes this is distribution. Again, this one is harder. Spend some time on it.