1. Prove the argument of the following statements using propositional or predicate logic Every living...

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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 a² -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 n² < 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 F₁ <2n for n²1 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 a² -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 n² < 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 F₁ <2n for n²1