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

Assignment 3 /AN