5. Let F = fAigi2I be an nonempty family of sets and let B be a set.

(a) B

T

i2I

Ai =

S

i2I

(B Ai):

(b) B

S

i2I

Ai

=

S

i2I

(B Ai):

(c) T

i2I

P (Ai) = P

T

i2I

Ai

:

6. Let F; G be nonempty families of sets such that F G. Prove:

(a) S

A2F

A

S

B2G

B:

(b) T

A2F

A

T

B2G

B:

7. Decide if each of the following relations is a well-deÖned mapping. If it

is not a mapping, state which of the two deÖning properties the relation

fails to possess.

(a) f : Q ! Q deÖned by f(

a

b

) = a

2

b

2 :

(b) f : Q ! Z given by f(

a

b

) = a + b:

(c) f : R

+ ! Z deÖned by f(n:d1d2d3:::) = d1:

(d) f = f(x; y) 2 R

2

: y = x

2g:

(e) f = f(x; y) 2 R

2

: x = y

2g:

8. Here are some examples from Calculus. Let C[0; 1] = f[0; 1] f! R : f is

continuousg: Which of the following maps are well-deÖned?

(a) E : C(I) ! R given by E(f(t)) = f(

1

2

):

(b) : C(I) ! R given by (f(t)) = lim

t!1

2

f(t): Is it true that E = ?

(c) : C(I) ! R given by (f(t)) = f

0

(

1

2

):

(d) : C(I) ! R given by (f(t)) = R 1

2

0

f(t) dt