## PrepTest 78, Game 1, Question 4

### Transcript

The fourth question of PT 78 Game 1 is a local question because we have a new piece of information. If T is not a project member, which one of the following works must be a project member? I'm gonna draw a new local diagram on my scratch paper to incorporate this new piece of information.

I'll go ahead and put T out, and then now, let's go to our rules. If T is out, rule number two, the contrapositive of it means that S also has to be out because if S were in, then T would also have to be in. But since T is now not in, S can be in, I'm going to throw S in the out group. Now at this point, it seems like we might be a little bit stuck because none of the other rules relate to T or S, so we're not quite sure where to go.

But let's go ahead and think about the different possibilities for rule one and rule three and that might help us figure out who has to be in. Starting with rule number one, remember if Q is in, Q' is got to be project leader, and if R is in, then R has to be project leader. And so if you try to put Q and R in at the same time if you try to do this, this is going to end up in a contradiction because each one wants to be project leader.

So since we can't put both Q and R in at the same time, what does that mean about at least one of Q or R? That means that at least one of these two letters has to be in our outgroup. That's why one of these outgroups spots right now actually is being taken up by Q or R. And I'm gonna draw Q/R in one of the outgroup slots to represent that.

There's no way that you're not going to see Q or R in the outgroup. Now let's take a look at rule number three which says that if W is selected in the in group, then R and V have to be out. So one possibility for this rule is, if you try to put W in, then we have to put R out, so I'll go ahead and circle R over here. And we also have to put V out and in this world, who are the remaining letters that would have to go in?

Well, you have the X and you also have the Q which would have to be the project leader. So that's what would happen if you put the W in. But let's go ahead and explore what would happen if you did not put the W in right now, because remember, the third rule only triggers if the W is in. Isn't it possible that you put the W out?

And once you put the W out, or outgroup is full, who are the remaining letters that would have to be forced in? Well, we have the X, you also have the V, and then you also have one of Q or R. We don't know which one but one of them would be in. So in both possibilities whether you put the W out, or whether you put the W in which remember it kicks out the R and V and the remaining letters are the X as well as the Q, in both possibilities whether W is in or W is out, the X has to be in at all times.

And that's why answer choice E is correct. If T is out at all times you do have to put an X in. Now I've just shown you the way that you can actually anticipate the answer here if you do the proper work. But we can also go ahead and get here through process of elimination. You can see the Q doesn't have to be selected because you could put the Q out and then you could put the R in.

That's also why answer choice B is wrong because you don't have to put R in you could put the Q in and then kick the R out. The V also doesn't have to be selected right now because you could put the V out as long as you put the W in and the R apps, and then the X and the Q in. You can also get rid of answer choice D, W, because if you put the W out that works as long as you put R in, X in and then, one of Q or R.

The X however does have to be in because if you tried to put X out, well now, we have to put W in and we also have to put the V in, these are two remaining letters and W and V can't go in together based on rule number three. So putting X out results in this contradiction, that's why X would have to be in.

Read full transcriptI'll go ahead and put T out, and then now, let's go to our rules. If T is out, rule number two, the contrapositive of it means that S also has to be out because if S were in, then T would also have to be in. But since T is now not in, S can be in, I'm going to throw S in the out group. Now at this point, it seems like we might be a little bit stuck because none of the other rules relate to T or S, so we're not quite sure where to go.

But let's go ahead and think about the different possibilities for rule one and rule three and that might help us figure out who has to be in. Starting with rule number one, remember if Q is in, Q' is got to be project leader, and if R is in, then R has to be project leader. And so if you try to put Q and R in at the same time if you try to do this, this is going to end up in a contradiction because each one wants to be project leader.

So since we can't put both Q and R in at the same time, what does that mean about at least one of Q or R? That means that at least one of these two letters has to be in our outgroup. That's why one of these outgroups spots right now actually is being taken up by Q or R. And I'm gonna draw Q/R in one of the outgroup slots to represent that.

There's no way that you're not going to see Q or R in the outgroup. Now let's take a look at rule number three which says that if W is selected in the in group, then R and V have to be out. So one possibility for this rule is, if you try to put W in, then we have to put R out, so I'll go ahead and circle R over here. And we also have to put V out and in this world, who are the remaining letters that would have to go in?

Well, you have the X and you also have the Q which would have to be the project leader. So that's what would happen if you put the W in. But let's go ahead and explore what would happen if you did not put the W in right now, because remember, the third rule only triggers if the W is in. Isn't it possible that you put the W out?

And once you put the W out, or outgroup is full, who are the remaining letters that would have to be forced in? Well, we have the X, you also have the V, and then you also have one of Q or R. We don't know which one but one of them would be in. So in both possibilities whether you put the W out, or whether you put the W in which remember it kicks out the R and V and the remaining letters are the X as well as the Q, in both possibilities whether W is in or W is out, the X has to be in at all times.

And that's why answer choice E is correct. If T is out at all times you do have to put an X in. Now I've just shown you the way that you can actually anticipate the answer here if you do the proper work. But we can also go ahead and get here through process of elimination. You can see the Q doesn't have to be selected because you could put the Q out and then you could put the R in.

That's also why answer choice B is wrong because you don't have to put R in you could put the Q in and then kick the R out. The V also doesn't have to be selected right now because you could put the V out as long as you put the W in and the R apps, and then the X and the Q in. You can also get rid of answer choice D, W, because if you put the W out that works as long as you put R in, X in and then, one of Q or R.

The X however does have to be in because if you tried to put X out, well now, we have to put W in and we also have to put the V in, these are two remaining letters and W and V can't go in together based on rule number three. So putting X out results in this contradiction, that's why X would have to be in.