Section 2.2
13b) Since A is a subset of A U B, every subset of A is a subset of A U B
Thus P(A) is a subset of P(A U B). Similarly, P(B) is a subset of
P(A U B). Thus, P(A) U P(B) is a subset of P(A U B)
13c) Let A={2} and B={1}. The P(A) U P(B)={{},{2}}U{{},{1}} while
P(A U B)={{},{1},{2},{1,2}}. In general, if A is a subset of B or B is
a subset of A, then P(A U B)=P(A) U P(B)
16b) Let B be contained in the Reals.
Then 0 is in B*, so B* is not empty.
16c) Suppose B=B*. Then it is clear that 0 is in B.
Next, suppose 0 is already in B then B* = B U {0} = B
16d) 0 is in B*, so (B*)* = B* by (c)
16e) (A U B)* = (A U B) U {0} by definition
= A U (B U {0}) = A U B*
But we can also write
(A U B)* = (B U A) U {0}
= B U (A U {0}) = B U A*
That is, we have A U B* = B U A*. Note that either way 0 is already in
the set. Applying part (c), we have (A U B)* = A* U B*
17b) Grade C. Need to state x belongs to A-C first, as well as x belongs to
B-C afterwards.
17c) Grade A
17f) This is a false claim.
Section 2.3
1)
f) union is {1,2,.
..,19},
intersection is empty
h) Union is [ -(pi), positive infinity)
Intersection is [ -(pi), 0]
j) Union is the set of all integers
Intersection is {0}
k) Union is (0,7/3)
Intersection is [1/3,2]
l) Union is (negative infinity, positive infinity)
Intersection is is { }
m) Union is R-Z (the set of all real numbers but not
include all the integers.)
Intersection is { }
n) Union is (-infinity, 1)
Intersection is (-1, 0]
2)