Math 320-101 CRN 22189 Spring 2016 Test 1 Carter 1. In an implication P ⇒ Q, which of P , Q is called the antecedent? Which is called the consequent? 2. Suppose that 0 < d and that d, n ∈ Z. Define d|n. 3. The statement, “Every non-empty set of natural numbers has a least element.” is called the . 4. Describe the two steps used to prove something by means of mathematical induction. . (a) First prove (b) Then prove . 5. Suppose that 3 divides both a and b. (a) Prove that for any integers x and y, the quantity xa + by is divisible by 3. (b) Prove that a · b is also divisible by 3. 6. Prove that any non-zero integer is divisible by itself. 7. Suppose that a, b, c ∈ Z. For convenience assume that a, b, c 6= 0. Suppose that a divides b and that b divides c. Show that a divides c. 8. Let r ∈ R be an arbitrary real number r 6= 1. Prove by induction that 1 + r + r2 + · · · + rn = (rn+1 − 1)/(r − 1). 9. Prove by induction that n X k=1 (2k − 1) = k 2 .