C Roettger, Spring 16
Math 201 assignment 2
Set theory, word problems
Please hand in by Wed 2/3 problems 2, 4, 6 from this handout, and from
the book problems I.6 # 6, I.7 # 10, I.8 # 10, II.2 # 6, 8, 12.
Problem 1 a) Shade in the region in a Venn diagram representing B ∩A.
b) Describe the shaded region of the Venn diagram below using union, intersection, and complement (∪, ∩, A).
A
B
A
B
Solution. For a),
For b), this is A ∪ B, or alternatively A ∩ B.
Problem 2 a) Shade in the region of the first Venn diagram representing
A ∩ B ∪ (A ∩ B).
b) Describe the shaded region of the second Venn diagram using union, intersection, and complement (∪, ∩, A).
A
B
A
B
Problem 3 a) Shade in the region of a Venn diagram representing
B ∩ A ∪ C.
b) Describe the shaded region of the Venn diagram below using union, intersection, and complement (∪, ∩, A).
C
A
B
Solution. For a),
C
A
B
For b), this is B ∩ A ∩ C.
Problem 4 a) Shade in the region of the first Venn diagram representing
A ∪ (B ∩ C).
b) Describe the shaded region of the second Venn diagram using union, intersection, and complement (∪, ∩, A).
C
C
A
B
A
B
Problem 5 You are investigating the shady organization Maple Alliance
For Industrial Advertising (M.A.F.I.A.). You have determined that 26% of
their profits come from Arms shipments, 21% of profits are connected to
Bribes paid to foreign officials , and 47% come from Clandestine operations.
Also, 16% of their profits are from clandestine arms shipments, 13% from
arms shipments connected to bribes. 11% of profits are from bribes paid
openly, not via clandestine.operations. Only 1% comes from arms sold openly
and without bribes.
Figure out the percentage of profits which goes in each region of the Venn
diagram for this situation.
In particular, which percentage of profits does not fit in any of these
categories?
Solution. From the last clue, you can put a 1 into the lower left region
of the Venn diagram. The sum of the other three parts of the set A is 26.
The top two of these sum to 16, the bottom two to 13, so they must be 12,
4, and 9 in this order. You can think of adding up all these numbers,
1 + x + y + z = 26
and you know
1 + (x + y) + (y + z) = 1 + 16 + 13 = 30.
So by counting y twice, we grow the sum by 4, which means y = 4, and the
rest follows like dominoes. The rest is then easy.
C
25
12
A
1
4
9
6
2
B
Problem 6 Suppose 20% of all bacteria in the ocean are anaerobic (A),
6% are bottom dwellers (B), and 17% are cyanobacteria (C).
Exactly half of all anaerobic bacteria are cyanobacteria, and half of the
bottom dwellers are cyanobacteria. The percentage of bacteria which are
cyanobacteria, but neither bottom dwellers nor anaerobic equals the percentage of bottom dwellers, and all anaerobic bottom dwellers are cyanobacteria.
Figure out the percentage of bacteria which goes in each region of the Venn
diagram for this situation.
In particular, which percentage of bacteria are anaerobic cyanobacteria,
but not bottom dwellers?