PRACTICE PROBLEMS 5

Let A = { a, b, c, d, e}, B = {a, c, q}, C = {e, f, g}.

1. Write each of the following.
a.  A Ç B
b. C -  (A Ç B)
c. (A  Ç C) È B
d. B ´ C
e. C ´ B
f.  B ´ B
g. The power set of C.

2. Let S = { a, b, 12, {Æ}, {{Æ}}, {Æ, Bill Gates}, {all positive integers less than 10}}. What is the cardinality of this set? How many elements are in the power set of this set?

3. From the universe of discourse U = {people}, let S = {All UMKC students} and R = {All persons born in Romania}. Using set-builder notation, describe  R Ç S, R  È S, and the complement of each of these.

Let A = {1, 2, 3} and B = {3, 4, 7}. Determine whether each of the following is:
    a relation or a function;
    If a relation or function, is it:
            One-to-One
            Onto
            One-to-One Correspondence
            Invertible
4. H = { (1, 3), (2, 3), (2, 7), (2, 4) }
5. I = { (1, 3), (2, 7), (3, 7) }
6. J = { (3, 7), (2, 3), (1, 4) }
7. K = { (3, 7), (7, 3), (1, 4), (4, 1), (1, 5)}

8. Let A = {Positive Integers < 10} and B = {Positive Integers < 15}. Define a relation between these sets which is:
a. Reflexive but not symmetric
b. Symmetric but not reflexive
c. Transitive but not reflexive
d. Antisymmetric but not reflexive
e. Neither symmetric nor antisymmetric.
f. One-to-one but not onto.
g. Onto but not one-to-one
h. Can there be a function between these sets which is both one-to-one and onto? If so, show one; if not, explain (i.e. prove) why not.

9. The characteristic function (of an item a for some set X) occurs frequently in computer science. It is denoted as CX (a) and defined as follows:

 CX (a) = 1 iff a is a member of set X;
CX (a) = 0 otherwise.
Prove that for any element a and any sets X and Y,
CX ÈY (a) = CX (a) + CY (a) - CXÇY(a)
Hint: Drawing a Venn diagram may help.

10 (based on a problem from Middleton). Let U be the set of all algebraic functions from the real numbers to the real numbers. Define a relation, denoted ~, between such functions as follows:

f(x) ~ g(x) iff | f(1) - g(1) | < 1
Determine whether the ~ relation is reflexive, symmetric, antisymmetric, transitive, an equivalence relation, or a partial order.
 
 

Answer Key

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