Exercises 145(e) "Some athletes are young females."
(f) "All athletes are young males."
(g) "Some athletes are female and are not young."
(h) "Some young females are not athletes."
(i) "All young females are athletes."(j) "Some athletes are not young."
(k) "No young people are athletes."
(1) "All athletes are either female or are young."
(m) "If all athletes are female, then all athletes are young; otherwise, no athletes are
young.'- Give an example of a universal set U and predicates P and Q such that (VxP(x))
(VxQ(x)) is true but Vx(P(x) --* Q(x)) is false. - Translate each of the following quantified formulas into an English sentence where the
universal set is R. Label each as true or false.
(a) Vx(3y(xy = x))
(b) Vy(3x(xy = x))
(c) Vx(3y(xy = 1))(d) 3y(Vx A O(xy = 1))
(e) 3x(Vy(xy = x))
(f) (Vx(x A 0-- 3y(xy = 1))
- Write a formula "saying" that at least four distinct objects satisfy predicate P.
- For any two integers m and n, we say m divides n if there is an integer k such that n =
ink. (Many programming languages give easy ways to say that, such as n % m == 0
or n div m = 0.) Define Div(m, n) to be m divides n. Translate each of the following
propositions and quantified formulas into a clear English sentence. Label each as being
true or false, with the universe as the set Z.
(a) Div(5, 7)
(b) Div(4, 16)
(c) Div(16, 4)(d) Div(-8, O)
(e) Vm(Vn(Div(m, n)))
(f) Vn(Div(1, n))
(g) Vm(Div(m, 0))
(h) Vm(Vn(Div(m, n) -- Div(n, m)))
(i) Vm(Vn(Vp((Div(m, n)ADiv(n, p)) -- Div(m, p))))
(j) Vm(Vn((Div(m, n)A Div(n, m)) -)i m = n))- Find a formula in negation normal form equivalent to the negation of
3xVyVz(P(x, y, z)).- Find formulas in negation normal form equivalent to the negations of each of the fol-
lowing:
(a) Vx(P(x) v Q(x))
(b) vx(Vy(P(x, y) - Q(x, y)))
(c) Vx((3yP(x, y)) - Q(x, y))
(d) Vx(((3y(P(x, y)) -+ Q(x, y)) A 3zR(x, z)))