Problem set 1 (b)Write a proposition equivalent to Ur P(r) using only the V quantifier, = and logical connectives Solution -Va(p(ar)v-vy(r=yvP(y)) Problem 3. A media tycoon has an idea for an all-news television network called lnn: The Logic News Network. Each segment will begin with the definition of some relevant sets and predicates. The days happenings can then be communicated concisely in logic notation. For example, a broadcast might begin as follows THIS IS LNN. Let S be the set Bill, Monica, Ken, Linda, Betty. Let D(ar)be a predicate that is true if a is deceitful. Let L(a, y)be a predicate that is true if r likes y. Let G(a, y)be a predicate that is true if r gave gifts to y. omplete the broadcast by translating the following statements into logic notation (a) If neither Monica nor Linda is deceitful, then Bill and Monica like each other Solution ((D(Monica)V D(Linda)))=>(L(Bill, Monica)A L(Monica, Bill)) (b) Everyone except for Ken likes Betty, and no one except Linda likes Ken Solution vr∈S(x=Ken∧-L(r, Betty)y(x≠Ken∧L(x,Bety)∧ vx∈S(x= Linda∧L(x,Ken)v(x≠ Linda∧=L(x,Ken) (c) If Ken is not deceitful, then Bill gave gifts to Monica, and Monica gave gifts to Solution D(Ken)→(G( Bill. monica)Ax∈SG( Monica,x) (d) Everyone likes someone and dislikes someone else Solution x∈S∈Sz∈S(y≠2)∧L(x,y)∧=L(x,x) The remaining problems will be graded primarily on the clarity of can't figure out the right idea, we'll give it to you-just ask your 1A, c your proofs- not on whether you have the right idea. In fact, if yo2 Problem Set 1 (b) Write a proposition equivalent to Ux P(x) using only the ∀ quantifier, =, and logical connectives. Solution. ¬∀x (¬P(x) ∨ ¬∀y (x = y ∨ ¬P(y))) Problem 3. A media tycoon has an idea for an all­news television network called LNN: The Logic News Network. Each segment will begin with the definition of some relevant sets and predicates. The day’s happenings can then be communicated concisely in logic notation. For example, a broadcast might begin as follows: “THIS IS LNN. Let S be the set {Bill, Monica, Ken, Linda, Betty}. Let D(x) be a predicate that is true if x is deceitful. Let L(x, y) be a predicate that is true if x likes y. Let G(x, y) be a predicate that is true if x gave gifts to y.” Complete the broadcast by translating the following statements into logic notation. (a) If neither Monica nor Linda is deceitful, then Bill and Monica like each other. Solution. (¬(D(Monica) ∨ D(Linda))) ⇒ (L(Bill, Monica) ∧ L(Monica, Bill)) (b) Everyone except for Ken likes Betty, and no one except Linda likes Ken. Solution. ∀x ∈ S (x = Ken ∧ ¬L(x, Betty)) ∨ (x = � Ken ∧ L(x, Betty)) ∧ ∀x ∈ S (x = Linda ∧ L(x, Ken)) ∨ (x = � Linda ∧ ¬L(x, Ken)) (c) If Ken is not deceitful, then Bill gave gifts to Monica, and Monica gave gifts to someone. Solution. ¬D(Ken) ⇒ (G(Bill, Monica) ∧ ∃x ∈ S G(Monica, x)) (d) Everyone likes someone and dislikes someone else. Solution. ∀x ∈ S ∃y ∈ S ∃z ∈ S (y �= z) ∧ L(x, y) ∧ ¬L(x, z) The remaining problems will be graded primarily on the clarity of your proofs— not on whether you have the right idea. In fact, if you can’t figure out the right idea, we’ll give it to you– just ask your TA!