m4 Let X be a set.(i) Show that the set of all subsets A of X suchthat A is finite or X\A is finite forms analgebra on X.(ii) Show that the set of all subsets A of Xsuch that A is countable or X \ A is countableforms a o-algebra on X.

Given Information:i) Set of all subsets of such that either is finite or is finite.ii) Set of…

Answered: Let X be a set. (i) Show that the set…

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 8TFE: True or False Label each of the following statements as either true or false. 8. A unity exists in...

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46. Let be a set of elements containing the unity, that satisfy all of the conditions in Definition a, except condition: Addition is commutative. Prove that condition must also hold. Definition a Definition of a Ring Suppose is a set in which a relation of equality, denoted by , and operations of addition and multiplication, denoted by and , respectively, are defined. Then is a ring (with respect to these operations) if the following conditions are satisfied: 1. is closed under addition: and imply . 2. Addition in is associative: for all in. 3. contains an additive identity: for all . 4. contains an additive inverse: For in, there exists in such that . 5. Addition in is commutative: for all in . 6. is closed under multiplication: and imply . 7. Multiplication in is associative: for all in. 8. Two distributive laws hold in: and for all in . The notation will be used interchageably with to indicate multiplication.
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]
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A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.
Exercises Work exercise 5 using U=a. Exercise5 Let U=a,b. Define addition and multiplication in P(U) by C+D=CD and CD=CD. Decide whether P(U) is a ring with respect to these operations. If it is not, state a condition in Definition 5.1a that fails to hold. Definition 5.1a: Suppose R is a set in which a relation of equality, denoted by =, and operations of addition and multiplication, denoted by + and , respectively, are defined. Then R is a ring with respect to these operation if the following conditions are satisfied : 1) R is closed under addition: xR,yRx+yR 2) Addition in R is associative: (x+y)+z=x+(y+z)x,y,zR 3) R contains an additive identity 0: x+0=0+x=xxR 4) R contains an additive inverse: for each x in R, there exists x in R such that x+(x)=(x)+x=0. 5) Addition in R is commutative: x+y=y+xx,yR 6) R is closed under multiplication: xR,yRxyR 7) Multiplication in R is associative: (xy)z=x(yz)x,y,zR 8) Two distributive laws holds in R: x(y+z)=xy+xz and (x+y)z=xz+yz x,y,zR

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Let X be a set. (i) Show that the set of all subsets A of X such that A is finite or X\A is finite forms an algebra on X. (ii) Show that the set of all subsets A of X such that A is countable or X \ A is countable forms a o-algebra on X.