Abstract Algebra Answers

In the set R of real numbers define an algebraic operation α by

α(a,b)=a-b

(a) (R,α) is not group

(b) α is not commutative

Let A={a,b,c} and P(A) the power set of A. List all the element of P(A). Show that the usual intersection, ∩ , and Union, U, of sets in P(A) are algebraic operation. What are the cayley's tables for there operation, Find the identity element if any, with respect to these operation.

Let π(N) be the number of primes less than or equal to N (example: π(100) = 25). The famous prime number theorem then states (with ∼ meaning asymptotically equal):
π(N) ∼ N/ log(N)
Proving this theorem is very hard. However, we can derive a statistical form of the prime number theorem. For this, we consider random primes which are generated as follows:
(i) Create a list of consecutive integers from 2 to N.

if a is an element of a group of finite order, prove that a^m ≠ a^n whenever m ≠ n

If M,N are R-module then M x N is also R-module

Let R be the ring and R^n= {(x1.......xn)/xi∈ R} be the R-module.

1] If I1,I2....In are ideal of R then N=I1xI2x....xIn={(x1.......xn)/xi∈ I} is a submodule in R^n

Any two disjoint permutation commute

|a|=(2*n+1);
aba^(-1)=b^(-1);
b^2=?

Use the fact that 0-a= -a to justify that
-(-5) equals 5.

*Question: Use the definition of negative numbers to justify that -(-5) equals 5.

Definition of negative numbers: -a is the number when added to a equals 0.

*Throw out and avoid the standard/traditional integer rules that you've learned and use only the knowledge we have gained working with negative numbers using the definition and number lines.
*This is a justification by proof type problem