Bridge to Abstract Mathematics: Mathematical Proof and Structures

110 LOGIC, PART II: THE PREDICATE CALCULUS Chapter 3

Then we define

(a) Prove that - [(Vx E A)(3y E B)p(x, y)] - (3x E A)(Vy E B)(-p(x, y)).

(b) Prove that - [(3x E A)(Vy E B)p(x, y)] t+ (Vx E A)(3y E B)(-p(x, y)). Note:

Parts (a) and (b) generalize, respectively, parts (a) and (b) of Theorem 3. It is

interesting to know that Theorem 1 generalizes also, namely

9. Translate each of the following symbolized statements into an English sentence,
   where I/ = R. Label each true or false:


10. Express in symbolic form each of the following English sentences, where we let
    U = R and recall that Z = set of all integers, Q = set of all rational numbers. In
    each case, decide also whether the statement is true or false:
    (a) There is a smallest real number.
    (b) There is no smallest real number.
    (c) There is an irrational number between any two reals.
    (d) There is a rational number between any two irrationals.
    (e) Every real number lies between two consecutive integers.
    (f) Every positive real number has a positive square root.
    (g) There exists a smallest rational number whose square is greater than 2.


11. Suppose p(x, y, z) and q(x, y) are propositional functions where each variable
    comes from a common nonempty domain of discourse U. One of the statement forms

is stronger than the other. Determine which is stronger by taking the negation

of both propositions and recalling the equivalence (p -+ q) t, (-q + -p). [Note:

• This exercise generalizes part (d) of Exercise 11, Article 3.3. This result is related
  to Exercise 4(b, c, d), Article 5.3.1

3.5 Analysis of Arguments for

Analysis of Arguments for Logical Validity, Part I1 (Optional)

In this article we consider methods of analyzing for logical validity argu-
ments whose partial premises and conclusion have any of the forms:
F.

It 1. All p's are q's.

2. Some p's are q's.