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Discrete Mathematics Assignment –

Q1) For A = {a, b, c, d, e} and B = {yellow, orange, blue, green, white, red, black}.
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?
Q2) Define functions f: R → R and g: R → R by f(a) = 2 + a and g(b) = 3b – 1. Find the following, showing the steps to get to your solution.
a) (f o g) (0).
b) (g o f) (1).
c) (f o g) (x).
d) (g o f) (x).
Q3) Let A = {CA, NH, IL, OH, SC, WV, PA, TX} and B = {book, table, chair, fork, road, car}. Using at least 5 ordered pairs, specify the following:
a) Define a function f from A to B that is one-to-one.
b) Define a function g from A to B that is not one-to-one.
c) Define a function h from A to B that is onto.
d) Define a function α from A to B that is not onto.
e) Define a function β from B to A that acts as the inverse of the function f that you created in part a) of this problem.
Q4) The function f: R → R defined by f(x) = 7x is onto because for any real number r, we have that r/7 is a real number and f(r/7) = r. Consider the same function defined on the integers g: Z → Z by g(n) = 7n. Explain why g is not onto Z and give one integer that g cannot output.
Q5) Let f: R → R be the function f(x) = x³ – 1. Find f^-1(x) and verify that it is the inverse of f.
Q6) Suppose a health insurance company identifies each member with an 8-digit account number. Define the hashing function h that first takes the first 3 digits of an account number as one number and the last 5 digits as another number, then adds them, and lastly applies the mod-37 function.
a) How many linked lists does this create?
b) Compute h(59243973).
c) Compute h(42280135).
Q7) Compute the check digit c for the 10-digit ISBN codes below. Show the calculations that you used to obtain your answers.
a) 0-523-76952-c (the initial 0 indicates that this is an English book).
b) 2-426-25967-c (the initial 2 indicates that this is a French book).
Q8) The picture below shows the graph of f(x) in red and the graph of b(x) in blue. Does the graph show that r is O(b), or that b is O(r), both, or neither? Explain your answer.

Q9) Define a relation R on the set of positive real numbers by (x, y) ∈ R if and only if x² – y² = 0. Determine if the relation R is a partial order. If it is not a partial order, explain which property or properties R fails to have.
Q10) Determine the ordered pairs in the relation determined by the Hasse diagram below on the set A = {a, b, c, d, e}. Create the matrix representation of this poset.

Q11) Define U = {1, 2, 3, 4, 5}. Consider the following subsets of U: A = {1, 2}, B = {3, 4, 5}, C = {1, 2, 5}, D = {5}
You may use (copy/paste/move/resize/etc.) the images below to create your graph.
a) Create the Hasse diagram using ⊆ as the partial order on the sets A, B, C, D, U, and •.
b) Is this a linear order? Explain your answer.

Q12) If < represents lexicographic order, then which of the following is/are true? Explain your answers.
a) (3, 11) < (3, 0)
b) (4, 7) < (2, 17)
c) (6, 2) and (8, 1) are not comparable because we need the first number to be larger in one of the pairs.
Q13) Let B = {2, 3, 4, 6, 12, 24, 36} 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. Explain your answers.