MAT 243 Online Written Homework Assignments for Week 2 (units 4-5)

(A natural number is a positive integer: 1,2, 3, etc.)

1. Formalize the following argument by using the given predicates and then rewriting the argument as a numbered sequence of statements. Identify each statement as either a premise, or a conclusion that follows according to a rule of inference from previous statements. In that case, state the rule of inference and refer by number to the previous statements that the rule of inference used. Dogs bark at cats. Max is a dog. Moonbeam is a cat. Therefore, Max barks at Moonbeam. Predicates: B(x,y)=x barks at y. D(x) = x is a dog. C(x) =x is a cat. The common domain of discourse is all animals.

2. Prove: the product of any two natural numbers is at least as big as either of them.

3. Prove that there exists a positive integer ???? that satisfies 2????+4=6. Make sure that you do not assume the conclusion. Assuming that a solution exists and deriving a value does not prove that a solution exists.

4. Prove that there does not exist a positive integer ???? that satisfies 2????+4=5. [Your proof must only use integers, inequalities and elementary logic. You may use that adding a number to an inequality, or multiplying an inequality by a positive number, preserves the inequality. There must be no reference to rational numbers or quotients.]

5. Prove: for every positive integer ????, there exists an integer ???? such that 1????+2<1????<1????. Make sure that you do not assume the conclusion.

6. Prove that for any two positive real numbers x and y, the arithmetic mean is at least as big as the geometric mean. The arithmetic mean of real numbers x and y is ????+????2. The geometric mean of two positive real numbers x and y is ????????. [You can use without proof that if a and b are positive quantities, and ????2????, then ????????.]