| 1. | (a) | Below is a partially completed table giving A u B,
A n B, |A u B|, and |A n B| for various
sets A and B. (Assume that a,b,c,... refer to distinct objects. Note
that I am using n and u as a rough approximation to the union and
intersection symbols due to limitations in web browser technology.
Complete the table. |
| |
| A |
B |
A u B |
A n B |
|A u B| |
|A n B| |
| {a,b,c,d,e} |
{b,c,d,e,f} |
{a,b,c,d,e,f} |
{b,c,d,e} |
6 |
4 |
| {a,b} |
{d} |
{a,b,d} |
Ø |
3 |
0 |
| {a,c,e} |
{c,d,e} |
|
|
|
|
| {a,b,f} |
{a,b,c,d,e,f} |
|
|
|
|
| {b,c,d,e} |
{c,f,g} |
|
|
|
|
|
| (b) |
Can you give a formula for |A u B| in terms
of |A|, |B|, and |A n B|? If so, explain why it is true. |
| |
| 2. | We discussed the power set P(S) of a set S in class. An
example of a power set is
P({black,white}) = { Ø, {black}, {white}, {black,white} }.
|
| (a) | List the elements of the power set of
{blue,red,yellow}. |
| (b) | What is the size of the power set of
{puce,turquoise,taupe,khaki,lime,cyan,emerald}? (You do not have to list
all the elements of the power
set, just give its size.) |
| |
| 3. | Write out the mathematical notation used as
shorthand for the following phrases. |
| (a) | x is not a member of the empty set. |
| (b) | The empty set is a subset of the powerset of U. |
| (c) | The intersection of A and B is equal to the union
of C and D. |
| (d) | The complement of X is a superset of Y. |
| (e) | X is the set of ordered pairs in which the first
item in the pair is taken from set Y and the second item is taken
from set Z. |
| |
| 4. | Let R be the relation between people and the cars
they own; that is, the pair (x,y) is in R if x is a person, y is a car,
and x owns y. Does R represent a function? Why or why not?
|