## LGB 2018 edition p. 73

Zarie Blackburn
PowerScore Staff

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We recently received the following question from a student:

"For question 11, I think the answer should be (F - G) -> F2, since the question says G cannot be cleaned UNTIL F is cleaned, unless F is cleaned second."
Jonathan Evans
PowerScore Staff

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Location: DFW, Texas

Hi!

Good question! First, there's not an error, but this is a tricky statement so let's look at it more closely. Start by using the Unless Equation™:

1. Whatever term is modified by “unless,” “except,” “until,” or “without” becomes the necessary condition.
2. The remaining term is negated and becomes the sufficient condition.

Let's use the equation here:

G cannot be cleaned until F is cleaned, unless F is cleaned second.

1. "F cleaned second" becomes the necessary condition.

F2

2. "G cannot be cleaned until F is cleaned." First, diagram this.

F G

Now, negate it.

G F

This becomes the sufficient condition. This is how we get:

(G F) F2

Let's discuss briefly what this statement means to reiterate how and why the Unless Equation works.

G cannot be cleaned until F is cleaned, unless F is cleaned second.

What does this mean? Our baseline scenario is one in which "G cannot be cleaned until F is cleaned" (F G). This is the way is has to be always, except for one scenario. In the event F is cleaned second (F2), then it is possible to have G before F.

Thus, what do we know? In all the scenarios in which F is not second, G is coming after F. This idea can be represented as follows:

F2 (F G)

Notice that this is the contrapositive of the conditional statement we made above using the Unless Equation:

(G F) F2

Remember that a conditional and its contrapositive are logically equivalent. Therefore, the explanation in the book is correct.

I hope this helps!