a.
Prove that the relation is an equivalence relation.
a.
Answer to Problem 6E
Equivalence Relation
Explanation of Solution
Given information:
The relation
Calculation:
Consider, the
Now check that the relation is reflexive, symmetric and transitive.
The relation is reflexive, symmetric and transitive.
Hence, this is an equivalence relation.
Equivalence class of:
b.
Find an element of
b.
Answer to Problem 6E
Equivalence Relation
Explanation of Solution
Given information:
The relation
Calculation:
Now check that the relation is reflexive, symmetric and transitive.
The relation is reflexive, symmetric and transitive.
Hence, this is an equivalence relation.
The element of
The element of
The element of
Three elements in the equivalence class
c.
Give the equivalence class of
c.
Answer to Problem 6E
Explanation of Solution
Given information:
The relation
Calculation:
Now check that the relation is reflexive, symmetric and transitive.
The relation is reflexive, symmetric and transitive.
Hence, this is an equivalence relation.
The equivalence class of :
d.
Name three elements in each of these classes:
d.
Answer to Problem 6E
Explanation of Solution
Given information:
On
Calculation:
Now check that the relation is reflexive, symmetric and transitive.
The relation is reflexive, symmetric and transitive.
Hence, this is an equivalence relation.
The equivalence class of :
e.
Describe the equivalence class of
e.
Answer to Problem 6E
Explanation of Solution
Given information:
The relation
Calculation:
Now check that the relation is reflexive, symmetric and transitive.
The relation is reflexive, symmetric and transitive.
Hence, this is an equivalence relation.
The equivalence class of :
f.
Find the number of elements in
f.
Answer to Problem 6E
Explanation of Solution
Given information:
For the set
Calculation:
Now check that the relation is reflexive, symmetric and transitive.
The relation is reflexive, symmetric and transitive.
Hence, this is an equivalence relation.
The equivalence class of :
g.
Describe all ordered pairs in the equivalence class of
g.
Answer to Problem 6E
Explanation of Solution
Given information:
The relation
Calculation:
Now check that the relation is reflexive, symmetric and transitive.
The relation is reflexive, symmetric and transitive.
Hence, this is an equivalence relation.
The equivalence class of :
h.
Name three elements in each of these classes:
h.
Answer to Problem 6E
Explanation of Solution
Given information:
Let
Calculation:
Now check that the relation is reflexive, symmetric and transitive.
The relation is reflexive, symmetric and transitive.
Hence, this is an equivalence relation.
The equivalence class of :
i.
Describe the equivalence class of
i.
Answer to Problem 6E
Explanation of Solution
Given information:
The relation
Calculation:
Now check that the relation is reflexive, symmetric and transitive.
The relation is reflexive, symmetric and transitive.
Hence, this is an equivalence relation.
The equivalence class of :
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Chapter 3 Solutions
A Transition to Advanced Mathematics
- 15. Prove that if for all in the group , then is abelian.arrow_forwardProve that the Cartesian product 24 is an abelian group with respect to the binary operation of addition as defined in Example 11. (Sec. 3.4,27b, Sec. 5.1,53,) Example 11. Consider the additive groups 2 and 4. To avoid any unnecessary confusion we write [ a ]2 and [ a ]4 to designate elements in 2 and 4, respectively. The Cartesian product of 2 and 4 can be expressed as 24={ ([ a ]2,[ b ]4)[ a ]22,[ b ]44 } Sec. 3.4,27b 27. Prove or disprove that each of the following groups with addition as defined in Exercises 52 of section 3.1 is cyclic. a. 23 b. 24 Sec. 5.1,53 53. Rework Exercise 52 with the direct sum 24.arrow_forward9. Find all homomorphic images of the octic group.arrow_forward
- Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)2=a2b2.arrow_forward24. Prove or disprove that every group of order is abelian.arrow_forwardLet G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.arrow_forward
- Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.arrow_forward13. Assume that are subgroups of the abelian group . Prove that if and only if is generated byarrow_forward16. Suppose that is an abelian group with respect to addition, with identity element Define a multiplication in by for all . Show that forms a ring with respect to these operations.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,