Please prove the following with clear steps and/or related theorems.Problem 2. If X is a set, P(X) denotes the set of all subsets of X. So for example, ifX = {x, y}, then P(X) is the set {Ø, {x}, {y}, {x, y}}. For subsets A and B of X, let A + Bdenote the set of x € X such that r is in both A and B or x is in neither A nor B.(i) Show that (P(X), →,U) is a ring.(ii) For subsets A and B of X, A®B denotes the set of points in X that belong to exactlyone of the sets A and B. Show that (P(X), , n) is isomorphic to (P(X), +,U).(iii) Show that every prime ideal of P(X) is maximal.

As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Please prove the following with clear steps and/or related theorems. Problem 2. If X is a set, P(X) denotes the set of all subsets of X. So for example, if X = {x, y}, then P(X) is the set {Ø, {x}, {y}, {x, y}}. For subsets A and B of X, let A + B denote the set of x € X such that x is in both A and B or æ is in neither A nor B. (i) Show that (P(X), →,U) is a ring.

Please prove the following with clear steps and/or related theorems. Problem 2. If X is a set, P(X) denotes the set of all subsets of X. So for example, if X = {x, y}, then P(X) is the set {Ø, {x}, {y}, {x, y}}. For subsets A and B of X, let A + B denote the set of x € X such that x is in both A and B or æ is in neither A nor B. (i) Show that (P(X), →,U) is a ring.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.4: Binary Operations

Problem 9E: 9. The definition of an even integer was stated in Section 1.2. Prove or disprove that the set of...

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9. The definition of an even integer was stated in Section 1.2. Prove or disprove that the set of all even integers is closed with respect to a. addition defined on . b. multiplication defined on .
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