Negative Integers Proof : Denote by -P the set of all negative integers, i.e., the set to which the number m belongs only in case there is a member n of P such that m = -n. If the number m is in -P..

Related BrainMass Solutions

• Sets and logic

  NOTE: The definition of a function being onto is: Suppose that f:X->Y is a function, f is onto if for every y in Y, there exists x in X such that f(x)=y So, if we want to prove a function which is onto, we have to show that for every y

• Proof using The Euler Function and Fermat's Theorem

  Proof The proof is completely analogous to that of the Fermat's Theorem except that instead of the set of non-negative remainders {1,2,...,m-1} we now consider the set {m1,m2,...,m (m)} of the remainders of division by m coprime to m.

• Determine the Nonnegative Integers

  ANSWER ALL PARTS OF THE QUESTION For every nonnegative integer n, there are unique nonnegative integers i and j such that and To see this, consider division of n by 43, and let i and j denote the quotient and the remainder, respectively

• The Distribution of Quadratic Residues

  Let N1(p) denote the number of pairs of integers in [1, p - 1] in which the first is a quadratic residue and the second is a quadratic nonresidue modulo p.

• Fundamental Theorem of Arithemtic : Lowest Common Multiples and Diophantine Equations

  This shows that there is no number in f(mIn) that is not in f(LCM(m,n)). To complete the proof, we need to show that there is no number in f(LCM(m,n)) that is not in f(mIN), which is simply done.

• Ruler function

  Let Observe that if then all rationals in have denominator greater than or equal to N. So, . Further since there are only m elements in R. Consequently, .

• Showing a Residue System Modulo to a Prime Number

  Equivalently, a complete residue system modulo m is a set satisfying (i) For every there exists an index i such that x = xi (mod m) (ii) xi xj (mod m) for i j. Proof.

• Many problems on group theory

  State the axioms for an equivalence relation ii. The relation n mod 3 divides the non-negative integers (i.e, n in Z such that n ≥ 0) into how many partitions? Show that n = 0 mod 3 is an equivalence relation. 2.

• Totient function proof

  (c) By definition, phi(p^i)= # {x : 1 =< x < p^i and x is coprime to p^i} Now, there are p^{i-1} - 1 integers less than p^i that are divisible by p, and these are the only ones having a common factor with p^i; in fact, they are p, 2p, ..., p(