Question

1. Suppose we have the following relation defined on Z. We say that a ∼ b iff 2 divides a + b. (a) Prove that the relation ∼ defines an equivalence relation on Z. (b) Describe the equivalence classes under ∼ .

2. Suppose we have the following relation defined on Z. We say that a ' b iff 3 divides a + b. It is simple to show that that the relation ' is symmetric, so we will leave that part off the assignment. However, the relation ' is not an equivalence relation. (a) Show that the relation ' is not reflexive. (b) Show that the relation ' is not transitive.

3. Consider the function f : R → [−1,∞) by f(x) = x 2 − 2x. (a) Prove that f is not one-to-one. (b) Prove that f is onto. Hint: the old quadratic formula and some basic reasoning about real vs complex numbers will be helpful here. (c) The ”function” g : R → [0,∞) by g(x) = x 2 − 2x has some problems. Describe why this information given about g fails to define a function. Give an example (there’s many ways) you might alter one of the domain or the rule for the function to define g so that it is a function.

4. Consider the function h : Z → Z by h(x) = 5x − 4. (a) Prove that h is one-to-one. (b) Prove that h is not onto.

5. The 10 digit ISBN number was developed in around 1970 and used as an international standard until around 2007 (when the number of digits were increased to 13.) The form of the 10 digit ISBN number is x1 − x2x3x4 − x5x6x7x8x9 − x10 where each of the first 9 are digits and the last is either a digit or an X representing the number 10. The first 9 digits are used for identification while the 10th digit is a ”check-digit” to make sure that the number was not improperly transcribed (or scanned.) The digits must satisfy the congruence below for the ISBN number to be valid 10x1 + 9x2 + 8x3 + 7x4 + 6x5 + 5x6 + 4x7 + 3x8 + 2x9 + x10 ≡ 0 (mod 11) (a) The following is an ISBN for a book on my table. 0-534-34028-x10. What is x10 and the title of the book? (b) The following is from a good book on my shelf and the x2 digit was scratched off. 0-x293-31604-1. What is x2 and the title of this book?

Answer #1

Let R be the relation on Z defined by:
For any a, b ∈ Z , aRb if and only if 4 | (a + 3b). (a) Prove that
R is an equivalence relation.
(b) Prove that for all integers a and b, aRb if and only if a ≡
b (mod 4)

Let's say we have the following relation defined on the set {0,
1, 2, 3}:
{ (0, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3) }
- Please answer the following 3 questions about this relation. (The
relation will be repeated for each question.) Is this relation a
function? Why or why not?
- What are the three properties that must be present in an
equivalence relation? Please give the names of the three properties...

For each of the following, prove that the relation is an
equivalence relation. Then give the information about the
equivalence classes, as specified.
a) The relation ∼ on R defined by x ∼ y iff x = y or xy = 2.
Explicitly find the equivalence classes [2], [3], [−4/5 ], and
[0]
b) The relation ∼ on R+ × R+ defined by (x, y) ∼ (u, v) iff x2v
= u2y. Explicitly find the equivalence classes [(5, 2)] and...

13. Let R be a relation on Z × Z be defined as (a, b) R (c, d)
if and only if a + d = b + c.
a. Prove that R is an equivalence relation on Z × Z.
b. Determine [(2, 3)].

1. A function f : Z → Z is defined by f(n) = 3n − 9.
(a) Determine f(C), where C is the set of odd integers.
(b) Determine f^−1 (D), where D = {6k : k ∈ Z}.
2. Two functions f : Z → Z and g : Z → Z are defined by f(n) =
2n^ 2+1 and g(n) = 1 − 2n. Find a formula for the function f ◦
g.
3. A function f :...

Recall from class that we defined the set of integers by
defining the equivalence relation ∼ on N × N by (a, b) ∼ (c, d) =⇒
a + d = c + b, and then took the integers to be equivalence classes
for this relation, i.e. Z = [(a, b)]∼ | (a, b) ∈ N × N . We then
proceeded to define 0Z = [(0, 0)]∼, 1Z = [(1, 0)]∼, − [(a, b)]∼ =
[(b, a)]∼, [(a, b)]∼...

3. For each of the piecewise-defined functions f, (i) determine
whether f is 1-1; (ii) determine whether f is onto. Prove your
answers.
(a) f : R → R by f(x) = x^2 if x ≥ 0, 2x if x < 0.
(b) f : Z → Z by f(n) = n + 1 if n is even, 2n if n is odd.

Determine the distance equivalence classes for the relation R is
defined on ℤ by a R b if |a - 2| = |b - 2|.
I had to prove it was an equivalence relation as well, but that
part was not hard. Just want to know if the logic and presentation
is sound for the last part:
8.48) A relation R is defined on ℤ by a R b if |a - 2| = |b -
2|. Prove that R...

2. Define a function f : Z → Z × Z by f(x) = (x 2 , −x).
(a) Find f(1), f(−7), and f(0).
(b) Is f injective (one-to-one)? If so, prove it; if not,
disprove with a counterexample.
(c) Is f surjective (onto)? If so, prove it; if not, disprove
with a counterexample.

Let S = {A, B, C, D, E, F, G, H, I, J} be the set consisting of
the following elements:
A = N, B = 2N , C = 2P(N) , D = [0, 1), E = ∅, F = Z × Z, G = {x
∈ N|x 2 + x < 2}, H = { 2 n 3 k |n, k ∈ N}, I = R \ Q, J =
R.
Consider the relation ∼ on S given...

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