Ex 1. (30 points) Diego, a bridge student, says: "In summer 2020 I will either go to Italy and eat an authentic Neapolitan pizza, or attend a Research Experiences for Undergraduates (REU) program in California." (1.1) Translate the above sentence into a propositional form, using upper case letters (specifying for each which simple proposition it represents), logical connectives and, if needed, paren- thesis. (1.2) Reading the future, we can see that in summer 2020 Diego went to Italy, he did not eat an authentic Neapolitan pizza and he attended a Research Erperiences for Undergraduates (REU) program in California. Was he lying? Justify fully your answer. Ex 2. (30 points) Consider the following propositions: Q:= "An icosahedron is a polyhedron with 20 faces". R:= "There are no real numbers between 3.14 and ". Determine the truth value of the propositional form ~P) AR) VQ), for the above propositions P, Q and R. Justify fully your answer. Ex 3. (40+10 points) (a) Consider the following definition: Definition A tautology is a propositional form that is true for every assignment of truth values to its components. Using a truth table, prove that the following propositional form is a tautology: (PAQ) ((P) V(Q)). (1) (b) Let A and B be two sets, and let : be an element of the union AUB. Let also: P:= " is an element of A". Q := "I is an element of B". Now answer the following questions: (61) In terms of set theory, what is the propositional form (1) saying in this particular case? You may use a Venn diagram to support your answer. (62) Explain what it means for the propositional form (1) to be a tautology in this context