(Finite Collections of Sets: Fundamental Products). Let U={1,2,3,4,5,6,7,8,9} be the universal set, and let A={2,3,4,5,6},B={1,5,6,7,9},C={2,3,5,6,7}. (i) Find all the fundamental products
P
1
=A∩B∩C,
P
5
=A
c
∩B∩C,
P
2
=A∩B∩C
c
,
P
6
=A
c
∩B∩C
c
,
P
3
=A∩B
c
∩C,
P
7
=A
c
∩B
c
∩C,
P
4
=A∩B
c
∩C
c
,
P
8
=A
c
∩B
c
∩C
c
of the sets A,B,C. (ii) Verify your answers by checking that the sets P
i
are pairwise disjoint and their union is equal to U. It is allowed not to include the solution of the following problem into your submission. However, the result may be of considerable help when you will work on Problem 3, and so please try to prove it, and keep the proof for your records. If you are going to include the proof of the result in your submission, please make sure that the proof contains all necessary details and is typed correctly. 2. (Counting Elements of Finite Sets). Let A,B,C be finite sets. Show that ∣A∪B∪C∣=∣A∣+∣B∣+∣C∣−∣A∩B∣−∣A∩C∣−∣B∩C∣+∣A∩B∩C∣. 3. (Counting Elements of Finite Sets). (i) Find subsets A,B,C of the universal set U={1,2,…,12} such that ∣A∣=5,∣B∣=7,∣C∣=6,∣A∩B∣=3,∣A∩C∣=3,∣B∩C∣=4 and ∣A∩B∩C∣=2 (here and below, any three subsets A,B,C of U satisfying the conditions will do). (ii) Find subsets A,B,C of the universal set U={1,2,…,15} such that ∣A∣=9,∣B∣=8,∣C∣=7,∣A∪B∣=12,∣A∪C∣=11,∣B∪C∣=11 and ∣A∪B∪C∣=13 (iii) Find subsets A,B,C of the universal set U={1,2,…,16} such that ∣A
c
∣=6,∣B
c
∣=11,∣C
c
∣=9,∣A
c
∪B
c
∣=13,∣A
c
∪C
c
∣=12,∣B
c
∪C
c
∣=12 and ∣A
c
∪B
c
∪C
c
∣=14, or prove that they do not exist (in the latter case please type the proof below the table with the other answers to the problem).