1. Prove the argument of the following statements using propositional or predicate logic Every living thing is a plant or animal. Jasmine's cat is alive, and it is not a plant. All animals have nose. Hence Jasmine's Children are smart. Smart people are revered. Nobody who can be illogical is smart. Children cannot be illogical. 2. Prove by contradiction that if a,b Z (a and b are integers) then a -4b +2 3. Prove by contrapositive for any integer a if 3a+1 is even then a is odd. 4. Prove by cases if a and b have opposite parties (i.e if a is even then b is odd and vice versa) then their sum is odd 5. Prove by mathematical induction the following. If n>=4 is an integer, prove by induction that n < n! Prove by mathematical induction that the sum of the cubes of three successive positive integers is divisible by 9. Prove by mathematical induction (check the nth expression of each sums) (a)the sum of the first n natural numbers (b) the sum of the squares of the first n natural numbers (c) the sum of the cubes of the n natural numbers. 6. Prove by strong induction that nth Fibonacci number F