Part 1:

1) Let A = {1, 2, 3, 4, 5} and B = {MA, NH, NV, TX, AK, ME}.

a) Define a relation R from A to B that is a function and contains at least 4 ordered pairs.
b) What is the domain of this function?
c) What is the range of this function?

2) Define functions f: R→R and g: R → R by f(a) = 5a-3 and g(b) = 4-2b. Find the following:

a) (f.g)(0)
b) (g.f)(1)
c) (f.g)(x)
d) (g.f)(x)

3) Let A = {2, 3, 5, 7, 11, 13, 17, 23} and B = {a, e, i, o, u, y}. Using at least 5 ordered pairs, define the following:

a) A function from A to B that is one-to-one.
b) A function from A to B that is not one-to-one.
c) A function from A to B that is onto.
d) A function from A to B that is not onto.
e) A function from B to A that acts as the inverse of the function you created in part a) of this problem.

4) The function f: R → R defined by f(x) = x³ is onto because for any real number r, we have that ³√r is a real number and f(³√r)=r. Consider the same function defined on the integers g: Z→ Z by g(n) = n³. Explain why g is not onto Z and give one integer that g cannot output.

5) Let A={x| x is a nation}
B = {Asia, Africa, North America, South America, Antarctica, Europe, Australia}

Let f:A→B be the function that outputs the continent to which a nation belongs. For example, f(Iceland) = Europe and f(Greenland) = North America. Explain why f is not a one-to-one function and give an example to prove it.

Part 2:

1) Suppose a health insurance company identifies each member with a 7-digit account number. Define the hashing function h which takes the first 3 digits of an account number as 1 number and the last 4 digits as another number; adds them, and then applies the mod-41 function.

a) How many linked lists does this create?
b) Compute h(4686158)
c) Compute h(9813284)

2) Compute the check digit c for the following ISBNs.

a) 031676948-c
b) 140123517-c

Part 3:

1) The picture below shows the graph of r(x) in red and the graph of b(x) in blue. Does this graph show that r is O(b) or that b is O(r)? Explain.

Part 4:

1) Define a relation R on the set of real numbers by (x,y) ε R if and only if x - y = 0. Determine if the relation R is a partial order. If it is not a partial order, explain which property or properties it fails to have.

2) Determine the ordered pairs in the relation determined by the Hasse diagram on the set A = {a, b, c, d, e}. Create the matrix representation of this poset.

3) Define U = {1, 2, 3, 4, 5}. Consider the following subsets of U:

P = {2, 3, 5}, O = {1, 3, 5}, E = {2, 4}, S = {3}

a) Create the Hasse diagram using ⊆ as the partial order on sets E, O, P, S, U, and Φ.

b) Is this a linear order? Explain.

4) If ? represents the lexicographic order, which of the following is/are true?

(5,12) < (5,4)
(5,4) < (5,12)
(5,4) and (5,12) are not comparable because we need the first number to be smaller in one of the pairs.

Part 5:

1) Let B = {1, 2, 3, 6, 12, 18} and R be defined by xRy if and only if x|y.

a) Determine all minimal and all maximal elements of the poset.

b) Find all least and greatest elements of the poset.