a.
Prove that the given sets are denumerable.
a.
Explanation of Solution
Given information:
Calculation:
An infinite set is denumerable if it is equivalent to the set of natural number. We have given
Now define the function
Hence, proved.
b.
Prove that the given sets are denumerable.
b.
Explanation of Solution
Given information:
Calculation:
An infinite set is denumerable if it is equivalent to the set of natural number. We have given
Now define the function
Hence, proved.
c.
Prove that the given sets are denumerable.
c.
Explanation of Solution
Given information:
Calculation:
An infinite set is denumerable if it is equivalent to the set of natural number. We have given
Now define the function
Hence, proved.
d.
Prove that the given sets are denumerable.
d.
Explanation of Solution
Given information:
Calculation:
An infinite set is denumerable if it is equivalent to the set of natural number. We have given
Now define the function
Hence, proved.
e.
Prove that the given sets are denumerable.
e.
Explanation of Solution
Given information:
Calculation:
An infinite set is denumerable if it is equivalent to the set of natural number. We have given
Now define the function
Hence, proved.
f.
Prove that the given sets are denumerable.
f.
Explanation of Solution
Given information:
Calculation:
An infinite set is denumerable if it is equivalent to the set of natural number. We have given
Now define the function
Hence, proved.
g.
Prove that the given sets are denumerable.
g.
Explanation of Solution
Given information:
Calculation:
An infinite set is denumerable if it is equivalent to the set of natural number. We have given
Now define the function
Hence, proved.
h.
Prove that the given sets are denumerable.
h.
Explanation of Solution
Given information:
Calculation:
An infinite set is denumerable if it is equivalent to the set of natural number. We have given
Now define the function
Hence, proved.
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Chapter 5 Solutions
A Transition to Advanced Mathematics
- If a is an element of order m in a group G and ak=e, prove that m divides k.arrow_forwardLet A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)arrow_forwardConstruct a multiplication table for the group D5 of rigid motions of a regular pentagon with vertices 1,2,3,4,5.arrow_forward
- 45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )arrow_forward24. Let be a group and its center. Prove or disprove that if is in, then and are in.arrow_forwardShow that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.arrow_forward
- Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .arrow_forwardIn Exercises 3 and 4, let be the octic group in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let be the subgroup of the octic group . Find the distinct left cosets of in , write out their elements, partition into left cosets of , and give . Find the distinct right cosets of in , write out their elements, and partition into right cosets of . Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group of rigid motions of a square The elements of the group are as follows: 1. the identity mapping 2. the counterclockwise rotation through about the center 3. the counterclockwise rotation through about the center 4. the counterclockwise rotation through about the center 5. the reflection about the horizontal line 6. the reflection about the diagonal 7. the reflection about the vertical line 8. the reflection about the diagonal . The dihedral group of rigid motions of the square is also known as the octic group. The multiplication table for is requested in Exercise 20 of this section.arrow_forwardIn Exercises 3 and 4, let G be the octic group D4=e,,2,3,,,, in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group D4 of rigid motions of a square The elements of the group D4 are as follows: 1. the identity mapping e=(1) 2. the counterclockwise rotation =(1,2,3,4) through 900 about the center O 3. the counterclockwise rotation 2=(1,3)(2,4) through 1800 about the center O 4. the counterclockwise rotation 3=(1,4,3,2) through 2700 about the center O 5. the reflection =(1,4)(2,3) about the horizontal line h 6. the reflection =(2,4) about the diagonal d1 7. the reflection =(1,2)(3,4) about the vertical line v 8. the reflection =(1,3) about the diagonal d2. The dihedral group D4=e,,2,3,,,, of rigid motions of the square is also known as the octic group. The multiplication table for D4 is requested in Exercise 20 of this section.arrow_forward
- 12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.arrow_forwardFor an integer n1, let G=Un, the group of units in n that is, the set of all [ a ] in n that have multiplicative inverses. Prove that Un is a group with respect to multiplication. (Sec. 3.5,3,6, Sec. 4.6,17). Find an isomorphism from the additive group 4={ [ 0 ]4,[ 1 ]4,[ 2 ]4,[ 3 ]4 } to the multiplicative group of units U5={ [ 1 ]5,[ 2 ]5,[ 3 ]5,[ 4 ]5 }5. Find an isomorphism from the additive group 6={ [ a ]6 } to the multiplicative group of units U7={ [ a ]77[ a ]7[ 0 ]7 }. Repeat Exercise 14 where G is the multiplicative group of units U20 and G is the cyclic group of order 4. That is, G={ [ 1 ],[ 3 ],[ 7 ],[ 9 ],[ 11 ],[ 13 ],[ 17 ],[ 19 ] }, G= a =e,a,a2,a3 Define :GG by ([ 1 ])=([ 11 ])=e ([ 3 ])=([ 13 ])=a ([ 9 ])=([ 19 ])=a2 ([ 7 ])=([ 17 ])=a3.arrow_forwardLet G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,