1. Four men and four women were nominated for two positions on
the school board. One man and one woman were selected to the
positions. Suppose the four men are named A, B, C, and D and the
women are named E, F, G, and H. Further, suppose that the following
statements are true: If neither C nor F won a position, then H won
a position; If either C won a position or G did not win a position,
then B won a position but F did not win a position. Who were the
two people elected to the school board and why?
2. Suppose a committee wants to decide the service award for
last year. There are a total of 6 candidates, Alice, Brigitte,
Chris, Dave, Emma, and Frank. Among the six candidates, Alice,
Brigitte, and Emma are females and the rest three are males. The
committee must decide exactly three people to be awarded, and the
nominations must meet all of the following criteria. At least one
female must be nominated; Chris and Emma cannot be both nominated;
Since Alice and Brigitte work together all the time last year, if
one of them is nominated, so is the other; Exactly one of the two
people Chris and Dave will be nominated; The nomination cannot be a
list containing exactly Alice, Brigitte, and Chris; If Chris cannot
get the nomination, neither can Brigitte. Now, as the chair of the
committee, how do you decide the nominations?
3. In the county jail, 4 prisoners A, B, C, D are selected and D
is blind. The sheriff Anne Oakley puts blindfolds on A, B, and C
since D is already blind. Next she selects four hats from seven
hats hanging on the hat rack, three of them are red and the other
four are white. She places four hats on the prisoner’s heads and
hides the remaining three hats. Then she takes the blindfolds off
A, B, and C and tells what she has done, including the fact that
there were three red hats and four white hats to choose from.
Sheriff Oakley then says, “If you can tell me the color of the hat
you are wearing, without looking at your own hat, you can go free.”
The following things happen: 1. A says that he can’t tell the color
of his hat. So the sheriff has him returned to his cell. 2. Then B
says that he can’t tell the color of his hat. So he is also
returned to his cell. 3. Then C says that he can’t tell the color
of his hat. So he is returned to his cell. 4. Then D, the blind
prisoner, says that he knows of the color of his hat. He tells the
sheriff, and she sets him free. What color was D’ hat, how did D do
his reasoning?
4. Five prisoners are arrested in a crime. But the jail is full
and the police has nowhere to place them. So he comes up with this
solution. He will give them a puzzle. If they succeed, all of them
go free; otherwise they are transferred to another jail in another
state. The police sets the 5 of them into a line and puts a hat on
each one of them. He explains to them that there are exactly 3
black hats and 3 white hats. The rule for the puzzle is simple. If
at least two of the prisoners can figure out the color of their own
hat in 45 seconds, all of them can go free. Prisoner A can see
prisoners B, C, D, and E. Prisoner B can see prisoners C, D, E.
Prisoner C can see prisoners D and E. Prisoner D can only see
prisoner E. Prisoner E cannot see anyone. And none of them can see
their own hat. After 15 seconds one of the prisoners blurts out “I
know the color of my 1 hat”. After another 15 seconds, another
prisoner blurts out the same words. Which two prisoners are they
and how do they find out?
5. Find a prenex normal form for the following wff. ∃x p(x) ∧ ∃x
q(x) → ∃x (p(x) ∨ q(x))