# first replacement 4 mp4

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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.