Sets:

Let A = {1, 2, ..., 8, 9}, B = {2, 4, 6, 8}, C = {1, 3, 5, 7, 9}, D = {3, 4, 5}, E = {3, 5}.

Which of these sets can equal a set X under each of the following conditions?

Conditions:

(a) X and B are disjoint : __________________________________

(b) (b) X D but X B : __________________________________

(c) X A but X C : __________________________________

(d) X C but X A : __________________________________

Relations:

1. Let S = {a, b, c}, T = {b, c, d}, and W = {a, d}. Find S T W.

3. Consider the relation R = {(1, 3), (1, 4), (3, 2), (3, 3), (3, 4)} on A = {1, 2, 3, 4}.

(a) Find the matrix MR of R.

(b) Find the domain and range of R.

(c) Find R 1.

(d) Draw the directed graph of R.

(e) Find the composition relation RR.

4. Let A = {1, 2, 3, 4}, B = {a, b, c} ,C = {x, y, z}. Consider the relations R from A to B and

S from B to C as follows: R = {(1, b), (3, a), (3, b), (4, c)} and S = {(a, y), (c, x), (a, z)}

(a) Draw the diagrams of R and S.

(b) Find the matrix of each relation R, S (composition) RS.

(c) Write R1 and the composition RS as sets of ordered pairs.

5. Let R and S be the following relations on B = {a, b, c, d}:

R = {(a, a), (a, c), (c, b), (c, d), (d, b)} and S = {(b, a), (c, c), (c, d), (d, a)}

Find the following composition relations:

(a) RS;

(b) SR;

(c) RR;

(d) SS.

Functions:

7. Let V = {1, 2, 3, 4}. For the following functions f: V V and g: V V ,

f = {(1, 3), (2, 1), (3, 4), (4, 3)} and g = {(1, 2), (2, 3), (3, 1), (4, 1)}

find:

(a) fg;

(b), gf ;

(c) f f :

8. Determine if each function is one-to-one.

(a) To each person on the earth assign the number which corresponds to his age.

(b) To each country in the world assign the latitude and longitude of its capital.

(c) To each book written by only one author assign the author.

(d) To each country in the world which has a prime minister assign its prime minister.

9. Let functions f, g, h from V = {1, 2, 3, 4} into V be defined by: f (n) = 6 n, g(n) = 3, h

= {(1, 2), (2, 3), (3, 4), (4, 1)}. Decide which functions are:

(a) one-to-one;

(b) onto;

(c) both;

(d) neither.