Question:Let B be a ring on 2, µ : B → R, be additive.Let A, BE B.(a) Show that u(A) = µ(A\ B) + µ(An B).(b) Showµ(A) + µ(B) = µ(AUB) + µ(An B).Hint: One way to do this is to use (a) and recall that (A\B)(B\A)(AUB) \(An B).(c)Show that 4(AUB) = µ(A)+µ(B) H(ANB) provided that µ(ANB) ER.If N is a nonempty set and BC P(N) is nonempty, then u : B → Ro iscalled a set function.Definition. Let B be a ring on N and µ : B → R• µ is called additive if µ(AUB) = µ(A) +µ(B) for all disjoint A, B e B.• u is called o-additive ifuH An) = EH(An)n=1n=1for any sequence (An) of pairwise disjoint sets in B with +, An E B.• µ is called non-negative if µ(A) > 0 for all A e B.• u is called increasing if µ(A) < µ(B) whenever A, B E B and A C B.• u is called subadditive if µ(A U B) < µ(A) + µ(B) for all A, B E B.• µ is called continuous from below ifu U An)-lim μ (Α)n00n=1for any increasing sequence (An) in B with UnEN An E B.

Name: Question:Let B be a ring on 2, µ : B → R, be additive.Let A, BE B.(a)
Uploaded: 2024-03-20T06:38:58.000Z
Channel: Bartleby
Description: Question:Let B be a ring on 2, µ : B → R, be additive.Let A, BE B.(a) Show that u(A) = µ(A\ B) + µ(An B).(b) Showµ(A) + µ(B) = µ(AUB) + µ(An B).Hint: One way to do this is to use (a) and recall that (A\B)(B\A)(AUB) \(An B).(c)Show that 4(AUB) = µ(A)

Given Ω a non-empty set, the power set of a set Ω, β be a ring on Ω μ:β→R∞ is a set additive…

Math

Advanced Math

(a) Show that µ(A) = µ(A\B) + 4(AN B). (b) Show u(A) + µ(B) = µ(AUB) + µ(An B). Hint: One way to do this is to use (a) and recall that (A\B)(B\A) = (AUB) \ (AN B). (c) Show that u(AUB) = µ(A)+µ(B)-u(AnB) provided that u(ANB) E R.

(a) Show that µ(A) = µ(A\B) + 4(AN B). (b) Show u(A) + µ(B) = µ(AUB) + µ(An B). Hint: One way to do this is to use (a) and recall that (A\B)(B\A) = (AUB) \ (AN B). (c) Show that u(AUB) = µ(A)+µ(B)-u(AnB) provided that u(ANB) E R.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.1: Definition Of A Ring

Problem 14E: Let R be a ring, and let x,y, and z be arbitrary elements of R. Complete the proof of Theorem 5.11...

See similar textbooks

Similar questions

Exercises Let I be a subset of ring R. Prove that I is an ideal of R if and only if I is nonempty and xy, xr, and rx are in I for all x and yI, rR.
19. Find a specific example of two elements and in a ring such that and .
Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .
Examples 5 and 6 of Section 5.1 showed that P(U) is a commutative ring with unity. In Exercises 4 and 5, let U={a,b}. Is P(U) a field? If not, find all nonzero elements that do not have multiplicative inverses. [Type here][Type here]
True or False Label each of the following statements as either true or false. Let x and y be elements in a ring R. If xy=0, then either x=0 or y=0.
Exercises Prove Theorem 5.3:A subset S of the ring R is a subring of R if and only if these conditions are satisfied: S is nonempty. xS and yS imply that x+y and xy are in S. xS implies xS.
a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].
Exercises 10. Prove Theorem 5.4:A subset of the ring is a subring of if and only if these conditions are satisfied: is nonempty. and imply that and are in .
Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4
Describe the kernel of epimorphism in Exercise 22. Assume that each of R and S is a commutative ring with unity and that :RS is an epimorphism from R to S. Let :R[ x ]S[ x ] be defined by, (a0+a1x++anxn)=(a0)+(a1)x++(an)xn.
Exercises If and are two ideals of the ring , prove that is an ideal of .
15. Let and be elements of a ring. Prove that the equation has a unique solution.