144 CHAPTER 2 Formal Logic
- For the following formulas, let the universe be R. Translate each of the following
sentences into a formula (using quantifiers):
(a) There is a smallest number.
(b) Every positive number has a square root. (Do not use the square root symbol; use
only multiplication.)
(c) Every positive number has a positive square root. (Again, do not use the square
root symbol; use only multiplication.) - For the following formulas, let the universe be R. Translate each of the following
sentences into a formula (using quantifiers):
(a) There is no largest number.
(b) There is no smallest positive number.
(c) Between any two distinct numbers, there is a third number not equal to either of
them. - Let U be the set of all problems on a comprehensive list of problems in science. Define
four predicates over U by:
P (x): x is a mathematics problem
Q(x): x is difficult (according to some well-defined criterion: it does not matter for us
what the criterion is)
R(x): x is easy (according to some well-defined criterion)
S(x): x is unsolvable (if you do not know what "unsolvable" means, do not worry
about it here)
Translate into English sentences each of the following formulas:
(a) VxP(x)
(b) 3xQ(x)
(c) Vx(Q(x) v R(x))
(d) Vx(S(x) --* P(x))(e) 3x(S(x) A -'P(x))
(f) -'(Vx(-'R(x) V S(x)))
(g) Vx(P(x) -- (Q(x) *" -R(x)))
(h) Vx-S(x)
(i) Vx(P(x) -- -S(x))
(j) Vx(P(x) -+ (R(x) V S(x)))
(k) 3x(-Q(x) A -R(x))
(1) 3x(R(x) A S(x))
(m) Vx(Q(x) -+ -R(x))- Let the universe U be the set of all human beings living in the year 2001, and translate
the following English sentences into quantified formulas. Let P(x) stand for "x is
young," Q(x) for "x is female," and R(x) for "x is an athlete."
(a) "All athletes are young."
(b) "Not all young people are athletes."
(c) "All young people are not athletes." (Warning: In informal English, this sentence
has two quite different meanings. One is "more grammatically correct" than the
other, however, and that is the one we're asking for.)
(d) "Some young people are not athletes."