Sets, Subsets and Venn Diagrams

1. (3 points)
- If A  B and B  C, can you conclude that A  C? Can you conclude that A  C?

Definitions of symbols:

⊂ "is a proper subset of"
⊆ "is a subset of"
⊄ "is not a subset of"

1) Yes. All of the elements in A are in B, and all of the elements in B are in C. Therefore, anything in A also has to be in C.

2) No. Think about the case of A = B = C.

- If A  B and B  C, can you conclude that A  C? Can you conclude that A  C?

1) Yes. A is a subset of B, B is a subset of C, therefore, A is a subset of C. However, C has some elements that are in not in B, and B has some elements that are not in A (each is a proper subset - it is a subset, but is not the same set - of the other set). This means that A is a proper subset of C (i.e. A ⊂ C).

2) Yes.

- If A  B and B  C, can you conclude that A  C? Can you conclude that A  C?

1) Yes.

2) Yes. The largest A can be is if A = B. But B is a proper subset of C, so A must also be a proper subset of C.

2. (4 points) Write down all possible subsets of {a, b, c, d}

Put the following in brackets to make them sets:

∅
a
b
c
d
ab
ac
ad
bc
bd
cd
abc
acd
abd
bcd
abcd

3. (2 points) Without writing them down what are the number of subsets of the set A = {a, b, c, d, e, f}? Of set B = {a, b, c, d, e, f, g, h, i, j}?

For A:

1 null set
6 sets with 1 element
15 sets with 2 elements
20 sets with 3 elements
15 sets with 4 elements
6 sets with 5 elements
1 sets with 6 elements

Total = 64.

In fact, the number of subsets of a set with n elements is 2^n. So, for A, the number of subsets is 2^6 = 64.

For B:

Number of subsets = 2^9 = 512.

4. (7 pts) Given U = {All letters of the alphabet} A = {b, c, d} and B = {c, e, f, g}
List the elements of set
(a) A U B (b) A ∩ B (c) A′ ∩ B′ (d) A′ U B′ (e) A U B′

Definitions:

∩ intersection
∪ union
Aor A' "the compliment of A"; all elements not in A
A - B all elements in A but not in B
n(A) "the number of elements in A"

a) Things in A and in B: {c}
b) Things in A or in B: {b, c, d, e, f, g}
c) Things not in A and not in B: whole alphabet except for {b, c, d, e, f, g}
d) Things not in A or not in B: whole alphabet except for {c}
e) Things in A or not in B: {a, b, c, d, h, I, j, k, ..., x, y, z}
f) Things in {a, b, c, d, h, I, j, k, ..., x, y, z} and in B {c}
g) Things in {c} and {a, b, c, d, h, I, j, k, ..., x, y, z} {c}

5. (2 points) Write the following in roster form: Set J is the set of natural numbers between one and seven.

{2, 3, 4, 5, 6}

(I'm assuming that "between" means greater than 1 and less than 7, but not including them.)

6. (2 points) State whether set A and B are equal, equivalent, both, or neither.
A = {9, 8, 10} B = {8, 9, 10}

Two sets are equal if they contain the same identical elements. These sets are equal because they each have 8, 9, and 10.

If two sets have only the same number of elements, then the two sets are equivalent (one-to-one correspondence). Equal sets are equivalent, but equivalent sets are not always equal sets.

Answer: both.

7. (2 points) Express the following in set-builder notation: M = {1, 2, 3, 4, 5}

{x | x is an integer and 1≤ x ≤ 5}

8. (8 pts) A drug company is considering manufacturing a new product that has two different flavors, orange and cherry. They surveyed 120 people. The results are as follows:
62 liked cherry flavor
74 liked orange flavor
35 liked both flavors.
Construct a Venn diagram and answer the following:
a) How many liked only orange flavor?
b) How many liked only cherry flavor?
c) How many liked either one or the other or both?

Draw two overlapping circles. In the overlap area, put the number 35 (number who liked both flavors). In the left circle, put the number 27 (62 - 35 = 27 people who liked just cherry). In the right circle, but the number 39 (74 - 35 = 39 who just liked orange).

This only accounts for 101 people (35 + 27 + 39 = 101). 101 people liked one or the other or both. Put the number 19 outside of both circles (19 people who don't like any).

9. (10 pts) In a survey of 75 resorts, it was reported that:
34 provided refrigerators in the guest rooms
30 provided laundry services
37 provided business centers
15 provided refrigerators in the guest rooms and laundry services
17 provided refrigerators in the guest rooms and business centers
19 provided laundry services and business centers
7 provided all three features.

Construct a Venn Diagram and use it to answer the following questions:
(a) How many of the resorts provided only refrigerators in the guest rooms?
(b) How many of the resorts provided exactly one of the features?
(c) How many of the resorts provided at least one of the features?
(d) How many of the resorts provided exactly two of the features?
(e) How many of the resorts provided none of the features?

I'll give you the numbers, you make the Venn Diagram (like in #9, but with 3 overlapping circles instead of 2). I calculated them from the bottom of the list (all three) to the top:

just refrigerators 34 - (8 + 7 + 10) = 9
just laundry 30 - (8 + 7 + 12) = 30 - 27 = 3
just business 37 - (10 + 7 + 12) = 37 - 29 = 8
fridge and laundry 15 - 7 = 8
business and laundry 19 - 7 = 12
fridge and business 17 - 7 = 10
all three 7

Total = 9 + 3 + 8 + 8 + 12 + 10 = 57.
18 resorts don't have any of the services.

a) only fridges 9
b) exactly one 9 + 3 + 8 = 20
c) at least one 57
d) exactly two 8 + 12 + 10 = 30
e) none 18