Math 523H Homework #3 Sections 1.4, 2.1 1.4 #1. Give 5 examples which show that P implies Q does not necessarily mean that Q implies P. If x R and x > 0, then x2 > 0. However, if x2 > 0, x is not necessarily greater than 0. If x R and x 0, then |x| 0. But
Math 523H Homework #2 Sections 1.2, 1.3 1.2 #1. Let S be the open interval (1, 2) and let T be the closed interval [2, 2]. Describe the following sets: (a) (b) (c) (d) (e) S T = [2, 2] = T S T = (1, 2) = S R\S = (, 1] [2, ) T \S = [2, 1] cfw_2 R\(T \S ) =
Math 523H Homework #1 Section 1.1 1.1 #1. Prove that if x and y are real numbers, then |xy | = |x|y |. Proof. Suppose that x, y R. Case 1: x < 0, y < 0. Then xy > 0, so |xy | = xy . But |x| = x and |y | = y , so |x|y | = (x)(y ) = xy by Prop 1.1.1(c). Thu