Given the universe U= {w, W2, W3} where: w is the world in which Sophia is a mathematician, Keith is a phonologist, and Lotus is a syntactician w is the world in which Sophia and Keith are pragmaticists and Lotus is i a syntactician w is the world in which Sophia is a sociolinguist, Lotus is a physicist, and Keith is a phonologist R is the relation between worlds in which if Sophia is linguist in the first one, then both Sophia and Lotus are linguists in the second one: R = { , < W, W >, < W, W3 >, < W, W >, < W3, W >}. i. Provide the truth conditions of the following propositions, i.e. the set of worlds in which the propositions are true (skill 2.1): a. There are at least three specialisations in this world. S = b. Sophia and Keith have the same specialisation. S = c. Sophia is a linguist. S3 = d. Sophia is a linguist or a mathematician. S4= ii. Perform the following operations on sets (skill 6.1): a. S S = b. S US3 = c. S U 82 d. S-S = e. S S = f. S = S = S = iii. Evaluate the following claims about sets as being true or false, and justify your answer (skill 6.2): a. {S} is a member of {S}. b. {S} is a subset of S. c. S is a proper subset of S. d. S is a subset of {S}. iv. Provide the property notation of the following set (skill 6.4): S5 = {W2, W3} v. Provide the list notation of the following set (skill 6.5): S6= {w | Lotus is a syntactician in wi} vii. Provide the characteristic functions of S3 and of R (shnfinkelized right- to-left) (skill 6.6). viii. Provide (list the elements of) the characteristic set of the following func- tions (the second is shnfinkelized right-to-left) (skill 6.7): W1 fSchar = W20 W30 fQchar = W10 Wi-> W1 W31 W0 W0 W30 W0 W3 W21 W31 W