1. For each of the statement below, decide whether it is true or false, and prove your claim. (a) For all a, b R, if a € Q \ {0} and b & Q then % & Q. (b) For all x, y € R, if x is irrational and y is positive and irrational, then x + √y is irrational.

If A is 3 x 3 and det (A) = 0, then select all of the following statements which are true:

Let Q be the set of all rational numbers. Which of these equations has no solution in Q? I: x? – 2 = 0 A. I and II only B. II only С. Ш only D. II and III only II: 2x2 +1 = 7 III: x3 – 4 = -31 E. None of the above.

Determine whether the statement is true or false. Prove the statement if it is true, and give a counterexample if it is false. (i) For all real numbers x and y, [x – y] = [x] – [y]. (ii) Let a, b and c be integers. If a|(bc), then a|b or a|c. (iii) Let m and n be integers. If m is even and n is odd, then m² – n² is odd.