Bridge to Abstract Mathematics: Mathematical Proof and Structures

58 LOGIC, PART I: THE PROPOSITIONAL CALCULUS Chapter 2

FTT T T F T

FFT T F F F

Figure 2.2 The statement form (-p A q) v (p A r) is true in some

cases, false in others.

Exercises

1. Which of the following are statements? (In some cases the best answer may be
   "a statement only if a specific context is understood."):
   (a) 4+3=7. (6) 5 - 7 > 0.
   (c) All men are mortal. (d) What day is today?
   (e) What a surprise! (f) sin2(3z/2) + cos2(3n/2).
   (g) sin2 x + cos2 x = 1. (h) sin2(0) + cos2(0) < 1.
   (i) Today is not Friday. (j) x2+6x+9=O.
   *(k) x2 + 6x +^9 = (x + 3)2. (I) f is continuous at a.
   (m) The day after Tuesday is Wednesday.
   (n) This book has a red cover.
   (0) Is there any real-valued function of a real variable that is continuous every-
   where and differentiable nowhere?
   / (p) If yesterday was Wednesday, then today is Thursday.
   /'
   (q) The baseball world series champions are in the American League.
   (r) The Baltimore Orioles are in the American League.
   (s) The president of the united States is a Republican.


2. (a) Set up an explicit one-to-one matching between the rows of a truth table
   for a statement form in n variables, say, p,, p,,... , p,, and the set of all subsets
   of the set {PI, P2,.. , P,}.
   (b) On the basis of (a) and Counting Formula 5(b), Article 1.5, how many dis-
   tinct rows are there in a truth table for a statement form in n variables?


3. Construct a truth table for each of the following statement forms: