Question 9. true or false? provide a simple counter-example if it is false. (9.a) the sum of two rational numbers x, y eq is always a rational number, x+y eq. (9.b) the sum of two irrational numbers x, y er - q is always an irrational number, x +y er-q. (9.c) the sum of the squares of two distinct real numbers x, y er, with x #y, is always a positive real number, x2 + y2 er and x2 + y2 > 0. (9.d) if a, b er, then ab e r.

a. True. Rational numbers are closed under the sum operation, therefore, the sum of two rational numbers is always a rational number.

b. True. irrational numbers are closed under the sum operation, therefore, the sum of two irrationals numbers is always a irrational number.

c. True. The square of a real number is always a number greater than zero, and the sum of two numbers greater than zero is greater than zero.

d. True. The real numbers are closed under the product operation, then if a and b are reals numbers, the product ab is also a real number.

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