Which of these relations on (0,1, 2, 3} are equivalencerelations?o {(0, 0), (0, 1). (0, 2), (1, 0), (1, 1), (1, 2), (2, 0). (2. 2). (3. 3)}o {(0, 0), (1, 1), (1, 3),(2, 2), (2, 3), (3, 1). (3, 2), (3, 3)}o {(0, 0). (0, 2), (2, 0), (2, 2), (2, 3). (3, 2), (3, 3)}• {(0, 0), (1, 1), (2, 2), (3, 3). (2, 3). (3, 2)}Clear my choice

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Answered: Which of these relations on (0, 1, 2,…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 11E: Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide...

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Prove Theorem 1.40: If is an equivalence relation on the nonempty set , then the distinct equivalence classes of form a partition of .
Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide whether or not is an equivalence relation. Justify your decision.
Prove that the equalities in Exercises hold for all in . Assume only the basic postulates for and those properties proved in this section. Subtraction is defined by .
In Exercises , prove the statements concerning the relation on the set of all integers. 18. If and , then .
Prove that the equalities in Exercises hold for all in . Assume only the basic postulates for and those properties proved in this section. Subtraction is defined by .
34. Let be the set of eight elements with identity element and noncommutative multiplication given by for all in (The circular order of multiplication is indicated by the diagram in Figure .) Given that is a group of order , write out the multiplication table for . This group is known as the quaternion group. (Sec. Sec. Sec. Sec. Sec. Sec. Sec. ) Sec. 22. Find the center for each of the following groups . a. in Exercise 34 of section 3.1. 32. Find the centralizer for each element in each of the following groups. a. The quaternion group in Exercise 34 of section 3.1 Sec. 2. Let be the quaternion group. List all cyclic subgroups of . Sec. 11. The following set of matrices , , , , , , forms a group with respect to matrix multiplication. Find an isomorphism from to the quaternion group. Sec. 8. Let be the quaternion group of units . Sec. 23. Find all subgroups of the quaternion group. Sec. 40. Find the commutator subgroup of each of the following groups. a. The quaternion group . Sec. 3. The quaternion group ; . 11. Find all homomorphic images of the quaternion group. 16. Repeat Exercise with the quaternion group , the Klein four group , and defined by
Exercises 33. Prove Theorem : Let be a permutation on with . The relation defined on by if and only if for some is an equivalence relation on .
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.
For determine which of the following relations onare mappings from to, and justify your answer. b. d. f.

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