Let R be an equivalence relation on domain D = {1,2,3,4} It is given that [1]=[2] What must be TRUE? a, [1] [3] b. R(1,3)→ R(3, 4) c. R(3,4) d. R(1,3)→ R(2, 3)
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![Question 42
Let R be an equivalence relation on domain D = {1,2,3,4}
It is given that [1]=[2]
What must be TRUE?
a, [1] [3]
b. R(1,3) → R(3, 4)
c. R(3, 4)
d. R(1,3) → R(2, 3)
a
b
C
d](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5bad4e48-dad8-4710-a64b-b24b80d1efcf%2Ffee85c77-92fd-49ed-818e-bb518b04377f%2Fd4v180v_processed.jpeg&w=3840&q=75)
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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 .7. a. Give an example of mappings and , where is onto, is one-to-one, and is not one-to-one. b. Give an example of mappings and , different from example , where is onto, is one-to-one, and is not onto.6. a. Give an example of mappings and , different from those in Example , where is one-to-one, is onto, and is not one-to-one. b. Give an example of mappings and , different from Example , where is one-to-one, is onto, and is not onto.
- 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.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
- Find mappings f,g and h of a set A into itself such that fg=hg and fh. Find mappings f,g and h of a set A into itself such that fg=fh and gh.Label each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.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 .