# Fundamentals of Mathematical Analysis by Rod Haggarty

-x +2 x -1 Solution Now 5x + 6 > 2x - 3 <::> 5x + 6 _ 2x - 3 > 0 x+2 x+2 x-1 <=> (Sx x-1 + 6)(x - 1) - (2x - 3)(x (x - l)(x + 2) + 2) > 0 3x 2 >0 (x-1)(x+2) <->--- Since x 2 ~ 0 for all x, the inequality is equivalent to (x - 1)(x + 2) > 0.

Material in this section will only be used when it is clear that it follows from the axioms. 1 1. Prove that \/5 is irrational and hence prove that u + h \15 is irrational for all rationals a and b, b =F 0. Deduce that the golden ratio r, defined by r = 1 + 1/r, r > 0, is irrational. 2. Which of the following statements are true? ::> x + y irrational. (b) x rational, y rational ~ x + y rational. (c) x irrational, y irrational=> x + y irrational. Prove the true ones and give a counterexample for each of the false ones.

Hence, for example, x = 0 and y = 2 will suffice. 2 1. Indicate on a diagram the following subsets of IR. x, y) : x 2 + y 2 oE; 2} H = {(x, y): x = 1} A C = {(x, y): y < 1} Hence sketch the sets An B, An C and <€AU C 2. Prove the follow:ing laws of the algebra of sets: (a) '€(~A) =A (b) An (Bu C) =(An B) u (A n C) (c) '€(A n B) =<€A u