Let A be nonempty set, and P(A) be the power set of A. Recall the definition Problem 2. of power set: P(A) = {r | x C A}. Show that symmetric deference operation on P(A) define by the formula x ® y = (x N y°) U (yna°), x € P(A), y E P(A), (where ye is the complement of y) the following statements are true: 1. The algebraic operation O is commutative and associative on P(A). Solution. 2. Show that for the operation e there exists an identity element. Solution. 3. Find all invertible elements Solution.

Let A be nonempty set, and P(A) be the power set of A. Recall the definition Problem 2. of power set: P(A) = {r | x C A}. Show that symmetric deference operation on P(A) define by the formula x ® y = (x N y°) U (yna°), x € P(A), y E P(A), (where ye is the complement of y) the following statements are true: 1. The algebraic operation O is commutative and associative on P(A). Solution. 2. Show that for the operation e there exists an identity element. Solution. 3. Find all invertible elements Solution.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 22E: A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which...

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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.