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https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/recitations/
Team Problem: A Mystery
A certain cabal within the 6.042 course staff is plotting to make the final exam ridiculously
hard. (“Problem 1. Prove that the axioms of mathematics are complete and consistent.
Express your answer in Mayan hieroglyphics.”) The only way to stop their evil plan is to
determine exactly who is in the cabal. The course staff consists of nine people:
{Oscar, Stav, Darren, Patrice, David, Nick, Martyna, Marten, Tom}
The cabal is a subset of these nine. A membership roster has been found and appears below,
but it is deviously encrypted in logic notation. The predicate incabal indicates who is in the
cabal; that is, incabal(x) is true if and only if x is a member. Translate each statement below
into English and deduce who is in the cabal.
incabal means in group
∃ Exists
!= Not Equal
∧ And
∨ Or
→ implies, if ..., then ..., IMPLIES
∀ for all
¬ Not
(i) ∃x ∃y ∃z (x != y ∧ x != z ∧ y != z ∧ incabal(x) ∧ incabal(y) ∧ incabal(z))
- 3 unique members exist (x, y, and z)
(ii) ¬ (incabal(Stav) ∧ incabal(David))
- Not Stav and David are in the group
- An equal meaning to this is either Stav OR David are in the group
(iii) (incabal(Martyna) ∨ incabal(Patrice)) → ∀x incabal(x)
- if Martyna OR Patrice are in the group then for all x are in the group.
- This is a lie because only 3 people can be in the group and for all x implies all 9
(iv) incabal(Stav) → incabal(David)
- if Stav is in, then David is in
- It just said above they both can't be in the group so Stav can't be in group
(v) incabal(Darren) → incabal(Martyna)
- if Darren is in then Martyna is in
- this leads to a dead end because of (iii)
(vi) (incabal(Oscar) ∨ incabal(Nick)) → ¬incabal(Tom)
- if Oscar or Nick then Tom is not in.
- So it could be Oscar and/or Nick
(vii) (incabal(Oscar) ∨ incabal(David)) → ¬incabal(Marten)
- if Oscar or David then not Marten
Therefore:
The only remaining people that could be in the group is:
Oscar, David, Nick
Team Problem: A Mystery
A certain cabal within the 6.042 course staff is plotting to make the final exam ridiculously
hard. (“Problem 1. Prove that the axioms of mathematics are complete and consistent.
Express your answer in Mayan hieroglyphics.”) The only way to stop their evil plan is to
determine exactly who is in the cabal. The course staff consists of nine people:
{Oscar, Stav, Darren, Patrice, David, Nick, Martyna, Marten, Tom}
The cabal is a subset of these nine. A membership roster has been found and appears below,
but it is deviously encrypted in logic notation. The predicate incabal indicates who is in the
cabal; that is, incabal(x) is true if and only if x is a member. Translate each statement below
into English and deduce who is in the cabal.
incabal means in group
∃ Exists
!= Not Equal
∧ And
∨ Or
→ implies, if ..., then ..., IMPLIES
∀ for all
¬ Not
(i) ∃x ∃y ∃z (x != y ∧ x != z ∧ y != z ∧ incabal(x) ∧ incabal(y) ∧ incabal(z))
- 3 unique members exist (x, y, and z)
(ii) ¬ (incabal(Stav) ∧ incabal(David))
- Not Stav and David are in the group
- An equal meaning to this is either Stav OR David are in the group
(iii) (incabal(Martyna) ∨ incabal(Patrice)) → ∀x incabal(x)
- if Martyna OR Patrice are in the group then for all x are in the group.
- This is a lie because only 3 people can be in the group and for all x implies all 9
(iv) incabal(Stav) → incabal(David)
- if Stav is in, then David is in
- It just said above they both can't be in the group so Stav can't be in group
(v) incabal(Darren) → incabal(Martyna)
- if Darren is in then Martyna is in
- this leads to a dead end because of (iii)
(vi) (incabal(Oscar) ∨ incabal(Nick)) → ¬incabal(Tom)
- if Oscar or Nick then Tom is not in.
- So it could be Oscar and/or Nick
(vii) (incabal(Oscar) ∨ incabal(David)) → ¬incabal(Marten)
- if Oscar or David then not Marten
Therefore:
The only remaining people that could be in the group is:
Oscar, David, Nick