By Eric Harold Neville

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Show that if a < 0, then a -1 < 0. 3. Given x > 0 and y > 0, show that x3 = y3x = y. Does this hold for every x and y? 4. Solve the following inequalities. The answers should be in one of the forms — <=> — or Ix 1 • • •}. = fx 1-1. 'Is it true that lx21 = 1x12 for every x? Why or why not? 6. Is it true that lx31 = 1x13 for every x? Why or why not? 7. Show that x2 — 2x + 1 0 for every x. 8. For what numbers x (if any) does each of the following conditions hold? 5 Order Relations and Ordered Fields 21 (c) lx — 51 = 12 x — 31 (d) 1x + 11 = 1 1— x1 (e) Vx 2 + 1 = x (f) 1/x 2— 1 = x (g) 12 • x — 11 + Ix + 31 13 x + 21 (h) 17 • x + 31 + 13 — xl >= 6Ix + 11 9.

If n 1, then n = k + 1, for some positive integer k. 4. ❑ ■ THEOREM 5. For each positive integer n, n + 1 is the smallest positive integer that is greater than n. PROOF. Let S be the set of all positive integers for which this holds. (1) 1 belongs to S. Proof: Suppose that there is a positive integer p such that p<1+1. 1< p > 1, it follows that p = k + 1 for some positive integer k. Thus p - 1 = k > 0. Therefore Since 0

Given PI, P2, ,P„, all different, such that no three of them are collinear and no four of them are coplanar. How many lines contain two of them? How many planes contain three of them? 5. Show that under our incidence postulates, S cannot be a line. 6. Show that there is at least one plane. 7. Show that there are at least two planes. 1 The Idea of a Function The word function is most commonly used in connection with calculus and its various elaborations, but the idea occurs, often without the word, in nearly all mathematics.

### Multilinear functions of direction and their uses in differential Geometry by Eric Harold Neville

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