1. Prove that if b and c are odd, then (a/bc)=(a/b)(a/c)

2. Prove that if a==b (mod c), where c is odd, then (a/c)=(b/c)

Solution Preview

For odd number P=p_1*p_2*...*p_s, where p_k are primes, the jacobi symbol (d/P) is defined as (d/P)=(d/p_1)(d/p_2)...(d/p_s), where (d/p_k) is the legendre symbol mod p_k.

1. If b and c are odd, ...

Solution Summary

Jacobi Symbols and Proofs are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

The problem is from Numerical Methods. Please show each step of your solution and tell me the theorems, definitions, etc. if you use any. Thank you.
Start with P0 = 0 and use Jacobi iteration to find.....
(Complete problem found in attachment)

Please help me learn how to write these two proofs correctly for my Modern Algebra class.
Please submit all work as either a PDF or MS Word file.
** Please see the attached file for the complete problem description **

Problem 1: Given the metric space (X, p), prove that
a) |p(x, z) - p(y, u)| < p(x, y) + p(z, u) (x, y, z, u is an element of X);
b) |p(x, y) - p(y, z)| < p(x, y) (x, y, z is an element of X).
These problems are from Metric Space. Please give formal proofs for both (a) and (b) based on the reference provided. Thank y

Please help with the following proofs. Answer true or false for each along with step by step proofs.
1) Prove that all integers a,b,p, with p>0 and q>0 that
((a+b) mod p)mod q = (a mod p) mod q + (b mod p) mod q
Or give a counterexample
2) prove for all integers a,b,p,q with p>0 and q>0 that
((a-b)mod p) mod q=0

(See attached file for full problem description with proper symbols)
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1A) Let R be a commutative ring and let A = {t R tp = 0R} where p is a fixed element of R. Prove that if k, m A and b R, then both k + m and kb are in A.
1B) Let R be a commutative ring and let b be a fixed ele

I need help with this program and maybe a small explanation:
Write a small program that will check for balancing symbols in C++. Look for:
•/* */
•( )
•[ ]
•{ }
See if you can display an error message that will describe the problem and point to the possible problem section of code.

Question: Why does abs (pi-4) = 4 - pi ? ( I think it's because pi -4 = -4+pi = -(4 - pi) and then abs (-(4 - pi)) = 4-pi, but I am not too sure). But here is what I really want to know, my book says abs (7 - pi) = 7- pi. If the above is true shouldn't it = pi - 7?

Please show these proofs in great detail with all steps explained as they will serve as a template for future proofs.
1. Suppose A, B, and C are sets with A?B?C = 0. Prove or disprove: |AUBUC|= |A|+|B|+|C|.
2. Suppose A, B, and C are sets. Prove or disprove: AUB= A?B if and only if A=B.