ICS 1F, Homework 2

Due Wednesday, January 28th, at the end of class

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?