Sample Problems for Midterm 1 The purpose of this document is to help you study for our first midterm. It is in no way Summary: Sample Problems for Midterm 1 The purpose of this document is to help you study for our first midterm. It is in no way a complete list of possible exam problems. In addition to completing these exercises, you should review your homework, notes and chapters 1,2,4,18,19,20, and 22. 1 Prove that if A and B are sets, then A B = A B A 2 Suppose that A is a set. Prove A 3 Suppose that f : A B and g : B C and that g f : A C is injective. Prove that f must be injective. 4 Express the following proposition using quantifiers: "Every real number can be approximated as closely as you like by a rational number." 5 Express the following proposition in plain English: x R, n N s.t. n > x. (This is called the Archimedean Principle) 6 Express the following proposition using quantifiers: "Every positive integer is the sum of two squares of integers". 7 Let a R \ Q and b Q \ {0}. Prove that ab R \ Q 8 Express the above proposition and its negation using quantifiers. 9 Disprove the following proposition (i.e. prove its negation): "Every odd integer is prime." 10 Disprove the following proposition: "The sum of any two irrational numbers is irra- tional." Collections: Mathematics