1. [4 Points] Let x be a positive real number. Prove that if m is irrational (i.e., not a rational1
number), then ﬂ is also irrational. For full credit your proof should be simple. 2. [4 Points] Additional Exercise 2.5.1(a). (Proof by cases.) Prove the statement: If a: is an
integer, then 1:2 + 5:1: — 1 is odd. 3. [6 Points] (Proof by cases) A perfect square is an integer n of the form n = m2 for some
integer m. Prove that every odd perfect square is of the form 8]»: + 1 for some integer k. 4. [8 Points] Consider the sets A = {2,6,7}, B = {2,4,9}, 0 = {1,2,4,5,7}. Use the
listing method (including commas and braces) to describe the following sets. (a) AUB
(b) 300
(c) AUBUC
(d) AﬂBﬂC
(e) (AUB)ﬂC
(f) AU(BﬂC)