Recitation 1 Team Problem: A Mystery A certain cabal within the 6.042 course staff is plotting to make the final exam ridiculous hard.("Problem 1. Derive all of known mathematics from first principles. Express your answer in Mayan hieroglypics The only way to stop their evil plan is to determine exactly who is in the cabal. The course staff consists of seven people Eric, Srini, Christos, Grant, Ishan, Sally, Theory Pig 1 (Sallyis the course secretary and Theory Pig is our mascot. )The cabal is a subset of these seven. A membership roster has been found and appears below, but it is deviously en crypted in logic notation. The predicate C indicates who is in the cabal; that is, C(a true if and only if a is a member. Translate each statement below into English and deduce who is in the cabal (1)x3y3(x≠y∧x≠zy≠z∧C(x)AC(y)AC(2) Solution. a direct English paraphrase would be"There exist people we'll call a, y, and z, who are all different, such that y and z are each in the cabal. " a better version would use the fact that there's no need in this case to give names to the people. Namely, a better paraphrase is"There are 3 different people in the cabal. Perhaps a simpler way to say this is: The cabal is of size at least 3 (ii)-(C(Theory Pig )AC(Grant ) Solution. Theory Pig and Grant are not both in the cabal. Equivalently: at least one of Theory Pig and grant is not in the cabal Solution. If Sallyis in the cabal, then everyone is (iv)C(Grant)-C(Theory Pig Solution. If grant is in the cabal, then Theory pig is also (v)(C(Ishan )V C(Christos ))-+-C(Srini Solution. If either of Ishan or Christos is in the cabal, then Srini is not. Equivalently, if Srini is in the cabal, the neither Christos nor Ishan is (vi)(C(Ishan )V C(Theory Pig ))--C (Eric Solution. If either of Ishan or Theory Pig is in the cabal, then Eric is not. Equiva lently, if Eric is in the cabal, the neither Ishan nor Theory Pig is sitions above is one whose members are exactly Theory Pig, Ishan, and Christl. propo- o much for the translations. We now argue that the only cabal satisfying all six� � � Recitation 1 2 Team Problem: A Mystery A certain cabal within the 6.042 course staff is plotting to make the final exam ridiculously hard. (“Problem 1. Derive all of known mathematics from first principles. Express your answer in Mayan hieroglypics.”) The only way to stop their evil plan is to determine exactly who is in the cabal. The course staff consists of seven people: {Eric , Srini , Christos , Grant ,Ishan , Sally, Theory Pig } (Sallyis the course secretary and Theory Pig is our mascot.) The cabal is a subset of these seven. A membership roster has been found and appears below, but it is deviously en￾crypted in logic notation. The predicate C indicates who is in the cabal; that is, C(x) is true if and only if x is a member. Translate each statement below into English and deduce who is in the cabal. (i) ∃x ∃y ∃z (x = y ∧ x = z ∧ y = z ∧ C(x) ∧ C(y) ∧ C(z)) Solution. A direct English paraphrase would be “There exist people we’ll call x, y, and z, who are all different, such that x, y and z are each in the cabal.” A better version would use the fact that there’s no need in this case to give names to the people. Namely, a better paraphrase is “There are 3 different people in the cabal.” Perhaps a simpler way to say this is: “The cabal is of size at least 3.” (ii) ¬(C(Theory Pig ) ∧ C(Grant )) Solution. Theory Pig and Grant are not both in the cabal. Equivalently: at least one of Theory Pig and Grant is not in the cabal. (iii) C(Sally) → ∀x C(x) Solution. If Sallyis in the cabal, then everyone is. (iv) C(Grant ) → C(Theory Pig ) Solution. If Grant is in the cabal, then Theory Pig is also. (v) (C(Ishan ) ∨ C(Christos )) → ¬C(Srini ) Solution. If either of Ishan or Christos is in the cabal, then Srini is not. Equivalently, if Srini is in the cabal, the neither Christos nor Ishan is. (vi) (C(Ishan ) ∨ C(Theory Pig )) → ¬C(Eric ) Solution. If either of Ishan or Theory Pig is in the cabal, then Eric is not. Equiva￾lently, if Eric is in the cabal, the neither Ishan nor Theory Pig is. So much for the translations. We now argue that the only cabal satisfying all six propo￾sitions above is one whose members are exactly Theory Pig , Ishan , and Christos