Numerical Methods and Optimization

Preliminaries 33

(b) f

−1

(Y



∩ Y



)=f

−1

(Y



) ∩ f

−1

(Y



);

(c) f(X



∪ X



)=f(X



) ∪ f(X



);

(d) f(X



∩ X



)maynotbeequaltof(X



) ∩ f(X



).

1.2. Prove that the following sets are countable:

(a) the set of all odd integers;

(b) the set of all even integers;

(c) the set 2, 4, 8, 16,...,2

n

,... of powers of 2.

1.3. Show that

(a) every infinite subset of a countable set is countable;

(b) the union of a countable family of countable sets A

1

,A

2

,... is

countable;

(c) every infinite set has a countable subset.

1.4. Show that each of the following sets satisfies axioms from Definition 1.4

and thus is a linear space.

(a) IR

n

with operations of vector addition and scalar multiplication. ...