Use the tableau method to prove or find a counterexample to Tv where T = {pVgV r. (pVg), (qAr) P.pVgVr} = (9) V-(q + (PVT)) Use the tableau method to find all models of the theory T over the language P {p.9.,} T = {p +9,9+r. ( -s), TV-} Show that the following CNF formula S (in set representation) is unsatisfiable by finding a resolution refutation. Draw the resolution tree. s={{P,9,-T, 8}, {-P.7,8}, {-4.-T}, {P.-8}, {-P.-T}, {r}} Consider the following two claims about coloring prime numbers of size at most n with two colors. where n > 1 is given. () There is no monochromatic (ie, colored with the same color) triple of distinct primes a,b, cn such that a+b=2c. (ii) If 3 and 7 ure colored with the same color, then 5 and 11 must be colored with the same color (but possibly different from the color of 3 and 7) as well. (a) Show how to write, for a given n > 1, the claim ) as a proposition Pn over the language P= {plie N is a prime), where p, means that " is colored by the first color". Then express the claim (11) as a proposition wover P. (b) In the following, let n = 13. Using the propositions 13 and , write a theory T which is unsatisfiable, if an only if any coloring satisfying (1) satisfies (11) as well. (c) Convert the axions of T to CNF and write the resulting CNF theory in set representation. (a) Use the resolution method to show that T U {p} is not satisfiable, that is, we additionally assume that the number 3 is colored with the first color. Draw the refutation in the form of a resolution tree. Let T = {-E(I,I), E(I,y) Ely,r), P.) 4 (Ey)(E(r,y)), } be a theory over the language L = (E,P) with equality, where is a binary relation symbol, P is a unary relation symbol, and the axiom y expresses that "there are exactly three elements". (a) Find all models of the theory T (up to elementary equivalence). (b) Let = P(I) E(I,y) Py). Is the sentence (Vr) (Vy) valid/contradictory/independent in T? (Don't forget to fully justify your answers!) Use the tableau method to prove or find a counterexample to Tv where T = {pVgV r. (pVg), (qAr) P.pVgVr} = (9) V-(q + (PVT)) Use the tableau method to find all models of the theory T over the language P {p.9.,} T = {p +9,9+r. ( -s), TV-} Show that the following CNF formula S (in set representation) is unsatisfiable by finding a resolution refutation. Draw the resolution tree. s={{P,9,-T, 8}, {-P.7,8}, {-4.-T}, {P.-8}, {-P.-T}, {r}} Consider the following two claims about coloring prime numbers of size at most n with two colors. where n > 1 is given. () There is no monochromatic (ie, colored with the same color) triple of distinct primes a,b, cn such that a+b=2c. (ii) If 3 and 7 ure colored with the same color, then 5 and 11 must be colored with the same color (but possibly different from the color of 3 and 7) as well. (a) Show how to write, for a given n > 1, the claim ) as a proposition Pn over the language P= {plie N is a prime), where p, means that " is colored by the first color". Then express the claim (11) as a proposition wover P. (b) In the following, let n = 13. Using the propositions 13 and , write a theory T which is unsatisfiable, if an only if any coloring satisfying (1) satisfies (11) as well. (c) Convert the axions of T to CNF and write the resulting CNF theory in set representation. (a) Use the resolution method to show that T U {p} is not satisfiable, that is, we additionally assume that the number 3 is colored with the first color. Draw the refutation in the form of a resolution tree. Let T = {-E(I,I), E(I,y) Ely,r), P.) 4 (Ey)(E(r,y)), } be a theory over the language L = (E,P) with equality, where is a binary relation symbol, P is a unary relation symbol, and the axiom y expresses that "there are exactly three elements". (a) Find all models of the theory T (up to elementary equivalence). (b) Let = P(I) E(I,y) Py). Is the sentence (Vr) (Vy) valid/contradictory/independent in T? (Don't forget to fully justify your answers!)