Only QUESTIONS 4 All parts are needed to be done

The following symbols can be used, instead of the stock ones: & can be used, instead of A, V (capital) instead of v, => instead of instead of --> instead of instead of - (+) instead of instead of - : instead of 1 - For each following formula, find a model which the formula is true, and one for which the formula is false. a. V x Vy (P(x,y)- P(y,x)) b. VxP(x) y(Q(x,y))) c. Vx (A() Vy (B(x,y)) 2 - For each item, translate to a formula using the modeling where: The universe set is the set of days (the usual rotation of the earth every 24 hours ...) of 2020 (numbered as days of the month); Unary predicates are, with their respective modeling = Rain (x) = "x is, or was, or will be, or has been, rainy"; or "x made rain"; or any sentence relating a given day to rain. Sun (x) = "x is, or was, or will be, or has been, sunny"; or "x was sunny" or any sentence relating a given day to be sunny. Sunday (x) = x is Sunday, Mon (x) = x is Monday, Tue(x) = x is Tuesday, Wed(x) = x is wednesday, Thur (x) = x is a Thursday; Friday (x) = x is Friday, Saturday (x) = x is Saturday; Today (x) = x is the current day. The binary predicate of equality x=y . The functions Yesterday (x) = returns the day before x Tomorrow (x) = returns the day after x a. Some days are sunny and rainy. b. Yesterday was sunny. Therefore, tomorrow and sunday will be rainy. c. It rained monday and tuesday. d. There will not be rain in the next three days. e. Some days are not rainy. 3 - Repeat what was done in the previous question, considering the modeling where the universe set is the set of all things, the binary predicate of equality, and the following unary predicates: P(x) = x is a person ; T(x) = x is a period of time ; E(x,y) = x tricks y; B(x) = x is a fool; I( x ) = x unscrupulous; 1 a. Each unscrupulous tricks at least one fool. b. Some persons tricks other persons during some period of time. c. Each fool is tricked by every unscrupulous. d. Some persons trick others all the time. Each unscrupulous tricks at least one fool. e. 4 - Repeat what was done in the previous question, considering the modeling where the universe set is the set of people, and the predicates are as follows: J(x) = x is a Judge ; PH( x ) = x is a Pharmaceutical ; L(x) = x is a Lawyer; W(x) = x is a Woman; AD( x, y) = x admires y ; Equals( x, y) = x=y II a. No woman is, at the same time, a Lawyer and a Pharmaceutical. b. All Judges admires only Judges. C. All woman Lawyers admires some Judge. d. There are men that admires at least two woman Lawyers