Recitation 1 We first observe that by (i), there must be someone -either Theory Pig or Grant-who is not in the cabal. But if Flo were in the cabal, then by(iii), everyone would be. So we conclude by contradiction that: Sallyis not in the cabal contradicting(ii). So by again contradiction, we conclude v), Theory Pig would be too, Next observe that if grant was in the cabal, then by (iv Grant is not in the cabal Now suppose Srini is in the cabal. Then by (v), Ishan and Christos are not, and w already know Sallyand Grant are not, so only three remain who could be in the cabal namely, Srini, Theory Pig, and Eric. But by (i)the cabal must have at least three mem- bers, so it follows that the cabal must consist of exactly these three. This proves Lemma 1. If Srini is in the cabal, then Theory Pig and Eric are in the cabal But by (vi), if Theory Pig is the cabal, then Eric is not. That is, Lemma 2. Theory Pig and eric cannot both be in the cabal Now from Lemma 2 we conclude that the conclusion of Lemma l is false. So by con- trapositive, the hypothesis of Lemma 1 must also be false, namely, Srini is not in the cabal Finally, suppose Eric is in the cabal. Then by(vi), Ishan and Theory Pig are not, and we already know Sally, Grant and Srini are not. So the cabel must consist of at most two people( Christos and Eric ) This contradicts (i), and we conclude by contradiction that Eric is not in the cabal o the only remaining people who could be in the cabal are Christos, Ishan, and Theory Pig ince the cabal must have at least three members we conclude that Lemma 3. The only possible cabal consists of Christos, Ishan, and Theory Pig But were not done yet: we havent shown that a cabal consisting of Christos, Ishan ind Theory Pig actually does satisfy all six conditions. So let A= Christos, Ishan, Theory Pig and let's quickly check that A satisfies (i)-(vi) .A=3, so A satisfies(i) grant is not in A, so A satisfies(ii)and (iv)Recitation 1 3 We first observe that by (ii), there must be someone – either Theory Pig or Grant – who is not in the cabal. But if Flo were in the cabal, then by (iii), everyone would be. So we conclude by contradiction that: Sallyis not in the cabal. (1) Next observe that if Grant was in the cabal, then by (iv), Theory Pig would be too, contradicting (ii). So by again contradiction, we conclude: Grant is not in the cabal. (2) Now suppose Srini is in the cabal. Then by (v), Ishan and Christos are not, and we already know Sallyand Grant are not, so only three remain who could be in the cabal, namely, Srini , Theory Pig , and Eric . But by (i) the cabal must have at least three mem￾bers, so it follows that the cabal must consist of exactly these three. This proves: Lemma 1. If Srini is in the cabal, then Theory Pig and Eric are in the cabal. But by (vi), if Theory Pig is the cabal, then Eric is not. That is, Lemma 2. Theory Pig and Eric cannot both be in the cabal. Now from Lemma 2 we conclude that the conclusion of Lemma 1 is false. So by con￾trapositive, the hypothesis of Lemma 1 must also be false, namely, Srini is not in the cabal. (3) Finally, suppose Eric is in the cabal. Then by (vi), Ishan and Theory Pig are not, and we already know Sally, Grant and Srini are not. So the cabel must consist of at most two people (Christos and Eric ). This contradicts (i), and we conclude by contradiction that Eric is not in the cabal. (4) So the only remaining people who could be in the cabal are Christos , Ishan , and Theory Pig . Since the cabal must have at least three members, we conclude that Lemma 3. The only possible cabal consists of Christos , Ishan , and Theory Pig . But we’re not done yet: we haven’t shown that a cabal consisting of Christos , Ishan , and Theory Pig actually does satisfy all six conditions. So let A = {Christos , Ishan , Theory Pig }, and let’s quickly check that A satisfies (i)–(vi): • |A| = 3, so A satisfies (i). • Grant is not in A, so A satisfies (ii) and (iv)