3.2 QUANTIFICATION 93Translate into an English sentence each of the following statement forms:- Let U be the set of all human beings living in the year 1987. Define three pred-
icates over U by:
dx): x is young.
q(x): x is male.
r(x): x is an athlete.
Express symbolically each of the statements:
All athletes are young. (b) Some athletes are not young.
Not all young people are athletes. (d) No young people are athletes.
All young people are not athletes.
Some young people are not athletes
Some athletes are young males. (h) All young males are athletes.
All athletes are young females.
Some athletes are male and are not young.
Some young males are not athletes.
All athletes are either female or are young.
Throughout Exercises 3 through 8, let U = Z, the set of all integers, and consider
pairs h(x) and k(x) of open sentences over Z given by:
(i) h(x): x is even k(x): x is odd
(ii) h(x): x is an integral multiple of^4 k(x): x is a multiple of^7(iii) h(x): x2 (^2 0) k(x): - 1 I sin x I 1
(iv) h(x): x2 I 0 k(x): Isin xl > 1
(v) h(x): x2 2 0 k(x): x is odd
(vi) h(x): x > 0 k(x): x < 0
/ 3. For & of the preceding pairs (i) through (vi), use Definition 1 in combination
with Definition 2, Article 3.1 [parts (b) and (c)] to label either true or false each of
the following statement forms. [Suggestion: First describe the truth sets H and K
for each of the six pairs. Then set up an 8 x 6 matrix with (a) through (h) as row
labels, (i) through (vi) as column headings, and Ts and F's as entries.]
(a) (Vx)(h(x) A k(x) (b) (Vx)(h(x) ) A (Vx)(k(x)
- (4 (Wh(x) A k(x)) *(a (W(h(x)) A (3x)(k(x))
(e) (W(h(x) v (Wk(4 f (Vx)(h(x) v k(x)
(91 (3x)(h(x)) v (3x)(k(x)) (h) (3x)(h(x) v k(x)
- In each of parts (a) through (h) of Exercise 3, translate the six symbolized state-
ment forms into English sentences corresponding to each of (i) through (vi). In each
case (a total of 48), compare your answer of true or false in Exercise 3 with your
intuitive judgment of truth or falsehood of your English translation.