Consider the following theorem: If r and s are rational numbers, then the product of r and s is a rational number (a) State the facts that we assume in a direct proof of this theorem. Be mathematically precise. (b) State the fact that we prove in a direct proof of this theorem. Again, be mathematically precise. 5. (10 points) Using a direct proof, prove the theorem stated in the previous question. 6. (10 points) Consider the following theorem: For any two real numbers, 2 and y, if x and y are both rational, then so is their sum. Rewrite this theorem as its contrapositive.

a) q ≠ 0 ,  b)  r .s = p1 / q1. p2 / q2  = p3/q3,  c)

Explanation:

A rational number is a number of the form p / q where the p and q values are integers,

Where we assume that q is different from zero

q ≠ 0

Since the division by zero is not defined

This is the only assumption to be made.

b) r .s = p1 / q1. p2 / q2

     If p1 and p2 are integers your product is another integer

         P1. p2 = p3

If q1 and q2 are integer your product is integer, none can be zero

        q1. q2 = q3

         p3 / q3 = r3

  What is a rational, what proves the theorem

c) the sum of two rational is another rational

           2 + y = r3

  Let's write the numbers with rational

          2 = p / q y = p2 / q2

           p / q + p2 / q2 = p3 / q3

             q = 1

             p = 2

            (2q2 + p2) / q2 = p3 / q3

We see that the numerator and denominator are true for which the theorem is true

answer; (d) spontaneous processes are characterized by the overall conversion of order to disorder;

as a general rule, scissors have blades less than 6 in (15 cm) long and usually have handles with finger holes of the same size. shears have blades scissors used to cut hair or fabric must be much sharper