1) Compute the Cayley tables for the additive group Z and for the multiplicative group
Z of non-zero elements in Z.

2) Let G be a group written additively. Recall that the order of an element a is the
minimal natural number n such that na = 0. If such n does not exits then one says that
the order of a is infinity.

i) Find the order of the following elements 2; 3; 5; 6 2 Z12.

ii) If G is a group written multiplicatively, the order of an element a is the minimal
natural number n such that an = 1. Find the order of the elements

iii) Find the order of the following elements...

3) i) Let G be a group written additively. An element a of a group G is called a generator
if any element x 2 G has the form x = na for some integer n. For example ????1 and 1 are
generators of Z, while Q has no generators at all. Find all generators of the group Z12.
ii) In multiplicative notation, an element a of a group G is called a generator if any
element of G can be written as a power of a. Carl Friedrich Gauss proved that for any
prime p the group Zp has a generator. Verify this statement for all primes 17 giving
explicitly a generator of the group Zp in each case.

Remark. Can you see any regularity among these generators for dierent primes? Probably not. A conjecture of Artin (which is still open) claims that if a is an integer which is
not a perfect square there are innitely many primes p for which a is a generator in Zp.

4) i) Let G be a group written multiplicatively. For any element a 2 G, consider the
map fa : G ! G given by fa(x) = ax. Prove that fa is always a bijection.

The response solves various problems in algebraic number theory, involving computing multiplicative orders of elements of Z_p and verifying bijective maps and quadratic reciprocity.

4 different math books, 2 different Chemistry books, and 6 different physics books are arranged on a shelf. How many different arrangements are possible if:
a.)The books in each particular subject must stand together?
b.)Only the math books must stand together?

Math.floor may be used to round a number to a specific decimal place. The statement
y = Math.floor( x * 10 + 0.5 ) / 10;
rounds x to the tenths position (i.e., the first position to the right of the decimal point).
The statement
y = Math.floor( x * 100 + 0.5 ) / 100;
rounds x to the hundredths position (i.e., the se

Please provide an explanation of Modular Arithmetic. I find it very confusing and am having trouble understanding the definitions and applying them.
If you could include some samples of some of the operations that might help.

Logic & Set Theory; Boolean Algebra; Relations & Functions
1. How do we distinguish relations from functions?
2. What sort of relation is friendship, using the human or sociological meaning of the word? Is it necessarily reflexive, symmetric, antisymmetric, or transitive? Explain why it is or is not any of these. What othe

Please help me solve this equation! Can you please also show me the steps as to how to solve this so I know in the future? Thank you
I was unable to put lines between the #'s so it's supposed to be 2y over y - 1 = 4 over y + 9 - 7y over y squared - y. as written below. Thank you
2y/(y - 1) = (4/y) + (9 - 7y)/(y^2

How do you decide how far to take math for students with intellectual disabilities? There are clear standards for students using regular education. So how do we choose standards for kids with intellectual disabilities?

Suppose F varies jointly as x and y and inversely as z squared. If F is equal to 18 when x = 4, y = 3 and z = 2, find the value of F if the values for x, y, and z respectively are doubled.

The A string on a string bass is tuned to vibrate at a fundamental frequency of 55.0 Hz. If the tension in the string were increased by a factor of four, what would be the new fundamental frequency?

A student had exams in math and Spanish. On the math the mean was u=30 and o= 5 and the student had a score of x=45. On the Spanish exam the mean was u= 60 and o=8 and the student has a score of x=68 . For which class can the student expect the better grade?
What is the z score corresponding to the mean?
What is the N val

Using the Fundamental Theorem of Calculus I need to find the solution of the following problems. Can you explain how?
Please see the attached file for the fully formatted problems.