Exercise 5.1.1
As a set:
speed 2.66 2.10 1.42 2.80 3.20 2.20 2.00 1.86 3.06
Average = 2.37
As a bag:
speed 2.66 2.10 1.42 2.80 3.20 3.20 2.20 2.20 2.00 2.80 1.86 2.80 3.06
Average = 2.48
Exercise 5.1.2
As a set:
hd 250 80 320 200 300 160
Average = 218
As a bag:
hd 250 250 80 250 250 320 200 250 250 300 160 160 80
Average = 215
Exercise 5.1.3a
As a set:
bore 15 16 14 18
As a bag:
bore 15 16 14 16 15 15 14 18
Exercise 5.1.3b
πbore(Ships Classes)
Exercise 5.1.4a
For bags:
On the left-hand side:
Given bags R and S where a tuple t appears n and m times respectively, the union of bags R and S will have tuple t appear n + m times. The further union of bag T with the tuple t appearing o times will have tuple t appear n + m + o times in the final result. On the right-hand side:
Given bags S and T where a tuple t appears m and o times respectively, the union of bags R and S will have tuple t appear m + o times. The further union of bag R with the tuple t appearing n times will have tuple t appear m + o + n times in the final result.
For sets:
This is a similar case when dealing with bags except the tuple t can only appear at most once in each set. The tuple t only appears in the result if all the sets have the tuple t. Otherwise, the tuple t will not appear in the result. Since we cannot have duplicates, the result only has at most one copy of the tuple t.
Exercise 5.1.4b
For bags:
On the left-hand side:
Given bags R and S where a tuple t appears n and m times respectively, the intersection of bags R and S will have tuple t appear min