Okay in this video we're going to do some logic word problems. This question
says, select a statement that is equivalent to the given statement which
is 'I ate a pickle or a hot dog.' If we think of this statement as a compound of
two propositions, 'I ate a pickle' being P and 'I ate a hotdog' being Q, then 'I ate a
pickle or a hotdog' is P or Q. So we converted that statement into a logic
statement and then we have multiple choice options so what we'll do is just
convert each of these multiple choice options into their logic statements,
create a truth table, and see which one is logically equivalent to P or Q. So a
says 'If I did not eat a pickle then I ate a hotdog.' That's if not P then Q.
'If I ate a pickle then I ate a hotdog.' That's if P then Q. 'If I did not eat a
pickle'--so not P--'then I did not eat a hotdog'--
so not Q. And d is 'If I ate a pickle then I did not eat a hotdog.' That's if P
then not Q. Okay so now we're ready to fill out our truth table. We have P, Q,
not P, not Q. Then we have P or Q, which is our given statement. We have not P
implies Q, P implies Q, not P implies not Q, and P implies not Q. Since we have
two propositions and each one can either be true or false, there will be four
possibilities for P and Q. We can do true, true, false, false. Then this will be
true, false, true, false to get a combination for all of them. Then not P
will just be false, false, true, true and not Q will be false, true, false, true. Then
P or Q will be true, true, true, false. Then with these implication statements,
if you have them in the form 'if P then Q,' if P is true and Q is true then this whole
statement is true. If P is true but Q is false, then this statement is false
because truth P is not implying Q in that case. If your first proposition is
false, then the implication could be anything so it could be false implies
true and that would be true or it could be false implying false and that whole
statement would still be true. Let's just do some examples to show you what I
mean. We have not P implies Q. We're
going to look at this column and the one next to it. Not P, the first row, is
false. Regardless of what Q is that's going to be true. The second one, it's the
same thing, then we have true, true. So that's true. Then we have true, false.
Truth is not implying truth so that whole statement has to be false and
we're done with that column. The next column--so that's going to be these two
columns--we have truth implying truth, truth implying false, so that's false and
then true, true and we're done there again. Then we have not P and not Q.
Since not P is false, false that will be true, true here and then true, false is
false and true, true is true. So we're done there and finally we have
P and not Q. We have true, false so that's false. True, true that's true,
and then true, true and we're done with our truth table. Now all that's left is
to compare this column, which is the column from the statement that we were
given 'I ate a pickle or a hot dog' and we're going to compare the values for
the other columns. The only other column that has the same truth values is the
one right next to it which is not P implies Q because for all instances of P
and Q these have the same values. Thus we can say P or Q is logically equivalent
to not P implies Q. So the statement not P
implies Q was a 'If I did not eat a pickle then I ate a hotdog' and we've
solved that problem. Okay so I was going to do another problem but as I'm looking
at it now I see it uses another technique besides truth tables. I
think I'll just leave that for another video and leave this as the only problem
in this video.