1.1

(a) False: a statement may be false.

(b) False: a statement cannot be both true and false.

(c) True: comment after Practice 1.4.

(d) True: comment before Example 1.3.

(e) False: if the statement is false, then its negation is true.

1.2

(8) False: p is the antecedent

(b) True: Practice 1.6(a).

(c) True: first paragraph on page 5.

(d) False: "p whenever q" is "if q, then p".

(e) False: the negation of pgis pa-q.

1.3 (a) M is not a cyclic subgroup.

(b) The interval [0,3] is not finite.

(c) Either the relation R is not reflexive or it is not symmetric.

(d) The set S is not finite and it is not denumerable.

(e) x3 and f(x) ≤7.

(f) fis continuous and A is connected, but f(x) is not connected. (g) K is compact and either K is not closed or K is not bounded.

1.4 (a) The relation R is not transitive.

(b) The set of rational mumbers is not bounded.

(c) Either the function is not injective or it is not surjective.

(d) x25 and x ≤7

(e) x is in 4 and f(x) is in B.

(f) f is continuous, but either f(x) is not closed or f(x) is not bounded.

(g) K is closed and bounded, but K is not compact.

1.5 (a) Antecedent: M has a zero eigenvalue; consequent: Mis singular.

(b) Antecedent: regularity; consequent: normality.

(c) Antecedent: it is Cauchy; consequent: a sequence is bounded.

(d) Antecedent: x = 5; consequent: f(x) = 14.

1.6 (a) Antecedent: 5n is odd; consequent: a is odd.

(b) Antecedent: it is monotone and bounded; consequent: a sequence is convergent.

(c) Antecedent; it is convergent; consequent: a real sequence is Cauchy. (

d) Antecedent: convergence; consequent: boundedness.

1.7 and 1.8 are routine.