Let p be an element in N, and define d _p to be the set of...

Let p be an element in N, and define d _p to be the set of all pairs (l,m) in N×N such that p divides m−l. Show that d_p is an equivalence relation

Expert Solution

Colby Messinger answered 2 years ago

7. Let n ∈ N with n > 1 and let P be the set of...

7. Let n ∈ N with n > 1 and let P be the set of polynomials with coefficients in R. (a) We define a relation, T, on P as follows: Let f, g ∈ P. Then we say f T g if f −g = c for some c ∈ R. Show that T is an equivalence relation on P. (b) Let R be the set of equivalence classes of P and let F : R → P be...

Let S = {a, b, c, d} and P(S) its power set. Define the minus binary...

Let S = {a, b, c, d} and P(S) its power set. Define the minus binary operation by A − B = {x ∈ S | x ∈ A but x /∈ B}. Show that (by counter-examples) this binary operation is not associative, and it does not have identity

The question is correct. Let X be an n-element set of positive integers each of whose...

The question is correct. Let X be an n-element set of positive integers each of whose elements is at most (2n - 2)/n. Use the pigeonhole principle to show that X has 2 distinct nonempty subsets A ≠ B with the property that the sum of the elements in A is equal to the sum of the elements in B.

Let X Geom(p). For positive integers n, k define P(X = n + k | X...

Let X Geom(p). For positive integers n, k define P(X = n + k | X > n) = P(X = n + k) / P(X > n) : Show that P(X = n + k | X > n) = P(X = k) and then briefly argue, in words, why this is true for geometric random variables.

Let A be a diagonalizable n × n matrix and let P be an invertible n...

Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP is the diagonal form of A. Prove that Ak = PBkP−1, where k is a positive integer. Use the result above to find the indicated power of A. A = 6 0 −4 7 −1 −4 6 0 −4 , A5 A5 =

Let A be a diagonalizable n × n matrix and let P be an invertible n...

Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP is the diagonal form of A. Prove that Ak = PBkP−1, where k is a positive integer. Use the result above to find A5 A = 4 0 −4 5 −1 −4 6 0 −6

Let G be a group, and let a ∈ G be a fixed element. Define a...

Let G be a group, and let a ∈ G be a fixed element. Define a function Φ : G → G by Φ(x) = ax−1a−1. Prove that Φ is an isomorphism is and only if the group G is abelian.

let D be an integral domain. prove that an element of D[x] is a unit if...

let D be an integral domain. prove that an element of D[x] is a unit if an only if it is a unit in D.

1)Show that a subset of a countable set is also countable. 2) Let P(n) be the...

1)Show that a subset of a countable set is also countable. 2) Let P(n) be the statement that 13 + 23 +· · ·+n3 =(n(n + 1)/2)2 for the positive integer n. a) What is the statement P(1)? b) Show that P(1) is true, completing the basis step of the proof. c) What is the inductive hypothesis? d) What do you need to prove in the inductive step? e) Complete the inductive step, identifying where you use the inductive hypothesis....

Let p be a prime and d a divisor of p-1. show that the d th...

Let p be a prime and d a divisor of p-1. show that the d th powers form a subgroup of U(Z/pZ) of order (p-1)/d. Calculate this subgroup for p=11, d=5; p=17,d=4 ;p=19,d=6

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Let p be an element in N, and define d _p to be the set of...

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7. Let n ∈ N with n > 1 and let P be the set of...

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