MATH 263 Towson University Discrete Mathematics Quiz Problems Discrete Mathematics. Solve problems using the definitions. Each problem uses the definition

MATH 263 Towson University Discrete Mathematics Quiz Problems Discrete Mathematics. Solve problems using the definitions. Each problem uses the definition of that property. Math 263 Discrete Mathematics
Dr Goode
Quiz #7
100 pts
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Work this Quiz with NO help from any other student or person. Only use methods and notation
seen in this particular course. Using unfamiliar notation looks like you are using an unauthorized
resource on the web. Most students do not have a library of reference books on mathematics. So
stick to what we have learned so you can transparently eliminate that factor in your general effort
to demonstrate that your work is your own.
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(0) Create a cover page for this quiz that is a whole piece of paper with the following written on it:
Your First and Last Name
Your original Section Number for this course (Sect: 020 or Sect: 021)
Quiz #7
Name your .pdf file as follows:LastName.FirstName.Quiz7.Sec02x where either x = 0 or x = 1, depending upon which
section you originally enrolled in.
Number your problems as indicated. Skip at least one line between consecutive problems and skip a line between
consecutive problem parts. Make sure your problems and pages are in order. Use only the definitions. Do not use any
prior result unless you proved it in an earlier problem in this quiz. Leave NOTHING to my imagination. Your reader should
not need to perform any calculations or make any inferences.
(1) (5 pts ea part) Let A = {a, e, g}, B = {c, d, e, f, g} and C = {j, k, m}.
(a) Construct a function f from A to B that is not one-to-one, if it is possible to do so. In that case, verify that your
subset f of A × B is a function, and explain exactly why it is not one-to-one. You need to explain how your specific function
f satisfies the definitions required to answer the questions here. If it is not possible to construct such a function, explain
exactly why it is not possible.
(b) Construct a function g from A to C that is a bijection, if it is possible to do so. In that case, verify that your subset
f of A × C is a function, and explain exactly why it is a bijection. You need to explain how your specific function g satisfies
the definitions required to answer the questions here. If it is not possible to construct such a function, explain exactly why it
is not possible.
(c) Construct a function h from C to A that is not onto, if it is possible to do so. In that case, verify that your subset h
of C × A is a function, and explain exactly why it is not onto. You need to explain how your specific function h satisfies the
definitions required to answer the questions here. If it is not possible to construct such a function, explain exactly why it is
not possible.
(d) Construct a function k from B to C that is a bijection, if it is possible to do so. In that case, verify that your subset
k of B × C is a function, and explain exactly why it is a bijection. You need to explain how your specific function k satisfies
the definitions required to answer the questions here. If it is not possible to construct such a function, explain exactly why it
is not possible.
(e) Draw the arrow diagram that represents your function f from part(a) above.
(f) Find f −1 , the set-theoretic inverse of your f from part(a) above.
(2) (4pts ea part) Let Let A = {a, e, g} and B = {c, d, e, f, g}. Let f : A → B and g : B → A be defined as follows:
f = {(a, c), (e, e), (g, d)}
g = {(c, a), (d, e), (e, e), (f, a), (g, g)}
(a) Consider the composed function g ◦ f .
(i) What is the domain of g ◦ f ? What is its codomain?
(ii) Find the function g ◦ f . (Find the set.)
(iii) Is g ◦ f one-to-one? Explain.
(iv) Is g ◦ f onto? Explain.
(b) Consider the composed function f ◦ g.
(i) What is the domain of f ◦ g? What is its codomain?
(ii) Find the function f ◦ g. (Find the set.)
(iii) Is f ◦ g one-to-one? Explain.
(iv) Is f ◦ g onto? Explain.
(3) (8pts) Use the Hamming Distance function to find H(100111, 001101).
(4) (10pts) Let the set S of data points be S = {762, 183, 367, 175}. Draw the Table that stores the data from set S when the
HASH function H : S → {0, 1, 2, 3, 4} that dictates the address for each data point is defined as follows:
H(n) = n mod 5.
Label the rows of your Table with the addresses (values from the codomain of the HASH function given here).
(5) (6 pts ea part) Consider the following function f : R → R:
f (x) = 4x + 5.
(a) Is f onto? If so, prove it. If not, give a counterexample to show exactly why not.
(b) Is f one-to-one? If so, prove it. If not, give a counterexample to show exactly why not.
(6) (8pts) Consider the set X = {0, 1, 2, 3, 4}. What is the identity function IX : X → X?
End Quiz #7

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