What is the maximum number of friends who could appear in a photo that Fiona does not appear in?

Question

An album contains photos picturing 7 friends: Al, Billy, Christina,Delores, Ed, Fiona, and Gloria. The friends appear either alone or in groups with one another, in accordance with the following: Ed appears in every photograph that Billy appears in. Billy appears in every photo that Delores appears in. Al appears in every photo that Fiona does not appear in. Neither Christina nor Al appear in any photo that Ed appears in. What is the maximum number of friends who could appear in a photo that Fiona does not appear in?

Answer 1

Let's number the statements to simplify things:

   (1) Ed appears in every photo the Billy appears in.

   (2) Billy appears in every photo that Delores appears in.

   (3) Al appears in every photo Fiona does NOT appear in.

   (4) Neither Christina nor Al appear in any photo that Ed appears in.

 

From (1)

   B --> E     (If Billy is in the photo, then Ed is in the photo)

From (2)

   D --> B     (If Delores is in the photo, then Billy is in the photo)

 

From (3) we are required to place Al in the photo.

From (4), this means that Ed cannot be in that photo (not E).

From the contrapositive of (1) this means that (not E) --> (not B). So Billy cannot be in that photo either (not B).

From the contrapositive of (2) this means that (not B) --> (not D). So Delores cannot be in that photo either.

 

Those are all the people that we can take out. So all the people that can appear in a photo without Fiona are Al, Christina, Ed, and Gloria. So your answer is 4 friends.