In this video, we're going to use truth tables to solve some logic problems. In this problem, we're
trying to see if this left-hand side is logically equivalent to the right-hand side and we can do
that using a truth table. If this has the same truth values for all propositions as this one,
then they are logically equivalent or if not, then this statement is false. One of the first
things you do when you're figuring out how many rows and columns you need for a truth table is
you figure out how many propositions you have. In this particular one, you have 3. You have P,
Q, and R and a proposition can be either true or false. P can be either true or false, so there are
two possibilities. Q can either be true or false and R can either be true or false. You're going
to have eight possibilities. There will be eight rows per column on your truth table. And I went
ahead and drew that ahead of time so the lines would be straight. If there were only P and Q,
then you would have four rows, and so on and so forth. The first three rows will just be our three
propositions and we already said that there will be eight rows. The easiest way to do it is just,
the first column will have half true and half false, and the easiest way will just be to do
the first four as true or false and then the last four as false in this case. These are not
going to be long enough. And then this one will be true true, false false, true true,
false false. And then this last column will always be true false, true false, true false, true false.
You don't have to do it that way but that's kind of the easiest way to remember how to make
sure you have a combination for all of them that's unique. And then the next column, you look for any
negations. We're going to have not Q and not R but there's no not P. We're going to put not Q, not R,
and we're basically just working to build up to get this statement and this statement in it's
entirety. Not Q, you just look at the Q column and put the opposite. This was true true, false false,
so you're going to put false false, true true, false false, true true. Then not R, you do the
same thing. It's going to be false true, false true, false true, false true, and now we have all
the individual letters. Next we can group letters. We have P or not Q and we're also going to have P
or not R. Let me go ahead and fill out the rest of the table in terms of the column headings. After
that, we'll have P or not Q and P or not R. Then we'll also have not Q and not R and we'll have P
or Q and not R. And now we can fill in the rest of the table. We're going to have P or not Q. We're
looking here and here. This column corresponds to this part of the logical statement. For this "or",
if we have P is true or not Q as true, then this whole statement will be true. This can be true
or false and this whole statement would still be true. The only way it could be false would be if
they were both false. For this column, we're going to look at P and we're going to look at the column
not Q that we already filled in. We see here that the way we wrote this, the first four are already
true for P. We can already say that this will be true true, true true, regardless of what not Q is.
False false, so that's false. False false again, so that's false again. And then these bottom two,
not Q are true here. We're going to get true and true. For this column, we're going to do the same
thing. This column refers to this part of the statement. Again, we're going to look at P and
we're going to look at not R, and if one of them is true then that whole statement will be true.
Again we know that the first four will be true because P are true for the first four. And then
not R, this is false, this is true. This is false, this is true. We have that column done. This next
column is the full statement on the left-hand side and this combines this statement. We're going to
look at this column and we're going to look at this column. For this one, this logical symbol is
"and", so both these columns have to be true for this whole statement to be true. If one is false,
then this whole statement is false. It's the opposite of this. These statements are both true,
so this is going to be true. Again it's true again, true true, false. We have an F for P or
not Q, so that's going to be false. We have one F here and we have two trues here, so that's true.
And now we're done with the left-hand side. Now we're going to work on this right-hand side. We
have not Q and not R, which is this part of the statement. We're going to look at not Q and not
R. And again because it's "and", both of these statements have to be true for the whole thing
to be true. We have false, false, false, true, false, false, false, true. And now we can fill in
the column for the whole right side, which will be looking at this column and P. You can see that the
truth table is basically building up left to right to get these more complex statements in a column.
Since we have "or" here, either P or this not Q and not R has to be true for the whole statement
to be true. Again we're going to know it's true, true, true, true. Then we're going to have false,
false, false, true. And now we're ready to see if they were logically equivalent
or not. The way we do that is we just focus on the columns that correspond to these two
statements. This is this column, P or not Q and P or not R. And this is this last column here,
P or not Q and not R. If they're the same for all the values, then this statement is logically
equivalent. If they're at all different at any row, then they're not equivalent. But we can see,
when this statement is true, this is true. This is true, true, true. When this is false, this is
false, so false, false, false. And then again we have true, true. You can see that this will always
imply this statement and this statement always implies that statement because they're always
logically equivalent. This statement is true and the two statements here are logically equivalent.