Need help with #1 and #8 especiallyNeed help with #1 and #8 especially

(5 points) a. Translate the following sentence into logic, using the provided predicates: "No one has been to every city in California" 1. B(x, y): x has been to y. Domain of x:All people, Domain of y:all cities C(z): z is in California. Domain of z: All cities b. Negate the logical statement, bringing the negation inside the quantifiers c. Translate the statement from b into English 2. (5 points) Prove that if x is irrational, then x is irrational 3. (5 points) Prove that if m, n are integers, and m+n is odd, then exactly one of m and n is odd. You MUST givea proof by contradiction (5 points) Show that the function f(x) -e from the set of real numbers to the set of real numbers is not invertible. Is it possible to change the sets the function acts upon in order to make it invertible? (2 points) (2 points) (5 points) positive integer 4. 5, 6, 7. Find the power set of the following set:, { }, a, b} Find the Cartesian product (AxB) of the sets: A = {a,b,c} B ={1,2 Use mathematical induction to show that 13 +33+53+ +(2n-1)3=(n+1)"(2n2+4n+1) whenever nisa 8. 2 points) There are 12 signs of the Zodiac. How many people must be in a group before you are guaranteed that at least 6 of them have the same sign? Refer to chapter 0 of the textbook, and below, for definitions Formal definitions: x is rational iff a.be Z x is irrational iff x is not rational x is even iff a e Z x is odd iff a E Z A function ffrom the set S to the set T is an assignment, for each element of S to an element in T. The functionf: S-> Tis one-to-one or injective iff ya S, beT(ua) = f(b)) a-b) The function/: S->Tis onto or surjective iff VteT 3s e S(f(s)- t) The functionf: S->Tis invertible iff fis both injective and surjective