Q-4: a) Let R and S be relations on the set A = {1,2,3} represented by the matrices [1 0 11 MR 1 i. List the elements of the set that represents the relation Rᵒ S. ii. Show that S is not transitive and find its transitive closure. [1 0 0 1 and Ms = 0 Lo 1 ol 1 01 0 0 0 1. b) Find the equivalence relation R on A with partitions {1,3,4}, {2}, {5}, and sketch its diagraph.

Q-4: a) Let R and S be relations on the set A = {1,2,3} represented by the matrices [1 0 11 MR 1 i. List the elements of the set that represents the relation Rᵒ S. ii. Show that S is not transitive and find its transitive closure. [1 0 0 1 and Ms = 0 Lo 1 ol 1 01 0 0 0 1. b) Find the equivalence relation R on A with partitions {1,3,4}, {2}, {5}, and sketch its diagraph.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.1: Finite Permutation Groups

Problem 33E: Exercises 33. Prove Theorem : Let be a permutation on with . The relation defined on by if and...

See similar textbooks

Similar questions

Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary elements of and ordered integral domain. If and, then. One and only one of the following statements is true: . Theorem 5.30 Properties of Suppose that is an ordered integral domain. The relation has the following properties, whereand are arbitrary elements of. If then. If and then. If and then. One and only one of the following statements is true: .
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 .
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
6. In Example 3 of section 3.1, find elements and of such that but . From Example 3 of section 3.1: and is a set of bijective functions defined on .
For any relation on the nonempty set, the inverse of is the relation defined by if and only if . Prove the following statements. is symmetric if and only if . is antisymmetric if and only if is a subset of . is asymmetric if and only if .
7. In Example 3 of Section 3.1, find elements and of such that . From Example 3 of section 3.1: andis a set of bijective functions defined on .
Prove Theorem 1.40: If is an equivalence relation on the nonempty set , then the distinct equivalence classes of form a partition of .
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 .
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.