1549901369-Elements_of_Real_Analysis__Denlinger_

604 Appendix A • Logic and Proofs

the original statement. Incorrect quantification can completely destroy the in-
tended meaning. Of special importance is the order in which we write multiple
quantifiers.

Example A.2.17 Consider the propositional function M(x, y) = "x is the
mother of y." There are six different statements that can be made by applying
quantifiers to this statement:

(a) Vx, Vy, M(x, y) = Everyone is mother to everyone.

(b) Vx,3y 3 M(x,y) =Everyone is a mother.

(c) 3x 3 Vy, M(x, y) =Someone is "mother-of-all."

(d) Vy, 3 x 3 M(x, y) = Everyone has a mother.

( e) 3 y 3 Vx, M ( x, y) = Someone has everyone as mother.

(f) 3 x, 3 y 3 M(x, y) = Someone is a mother.

Note that a slight change in the quantifiers used, or in the order in which they

are written, can have a drastic effect on the meaning of the statement. D

EXERCISE SET A.2

PART A: In Exercises 1- 28, define approp riate propositional functions,
specify the domain(s) of t he variable(s), and translate the given statement into
symbolic form.

Example: All unicorns have four legs and one horn.

Solution: Let the domain of x be the set of all animals. Let U(x) = "xis
a unicorn," L(x) = "x has four legs,'' and H(x) = "x has one horn." Then the
given proposition is symbolized:

Vx, [U(x):::;. {L(x) /\ H(x)}].

1. Lawyers are all wealthy.


2. Anyone who wants to succeed must work hard.


3. No one who wants an A in this course can afford to miss an assignment.
   (Two equivalent ways-see note following Table A.11.)


4. Someone in the class will win the raffle.


5. Someone in this room is guilty, but no one in this room will be charged.