Chapter2TruthTablesB

In this video, we will look at 2 examples

to help us practice drawing truth tables,

which, remember, are just a way to organize

possible truth values in a table format

when you have, sometimes, pretty complicated situations with p's and q's.

So when you're doing a truth table,

the first thing you want to do is figure out

how many columns that you're going to need.

In this case, we're drawing a truth table for p, q, and p and q.

So our ultimate end is going to be p and q,

and before we get there we're going to have to have

p individually and q individually.

So altogether you need 3 columns.

Now, we're working with 2 variables, so the first thing you need to do is figure out

all the possible combinations of truths for p and q, for 2 variables.

So they could both be true; you could have, also, 1 be true and 1 be false.

And there's 2 ways for that to happen, because either 1 could be true.

And finally, it could also be that they're both false.

So for 2 variables there's 4 possible truth combinations to start.

Once you've done that, you want to analyze p and q.

So for p and q to be true, remember, that's the symbol for and,

both p and q have to be true.

So if p is true and q is false, then p and q will be false.

So they both have to be true for p and q to be true.

So the only time where both of them are true is the first 1.

So in that case, p and q will be true,

but in the rest of them p and q would be false.

So that's a truth table for p, q, and p and q.

All right. Second example.

Draw a truth table for p, q, and p or q.

So the same thing, except this time we're doing 'or' instead of 'and.'

So again, we're going to have 3 columns,

for p and q each by themselves, and then p or q.

So again, once we've realized that we're working with 2 variables,

the first thing you're going to do is fill in all the possible truths

for p and q, for 2 variables, and there should be 4.

If there were 3 variables, there should be 8,

if there are 4 variables, there's 16.

They're actually always powers of 2, the number of possible combinations.

So for 2 possible variables.

They could both be true, you could have 1 be true and 1 be false,

or the other 1 be false and the other 1 be true,

or they could both be false.

Those are the 4 possible combinations.

Now, with this symbol, this is the symbol for or.

And for p or q to be true, it just means that at least 1 of them has to be true

but not necessarily both.

So p or q, you should look through and notice

in almost all of these, 1 of them is true.

There's only 1 case where neither of them are true, which is right here.

So that's the only case where our answer will be false,

but for all of these we're going to just have true, true, true.

So p or q is true most of the time, unless

the original 2 statements, both of them were false.