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.
Want to see more full solutions like this?
Chapter 5 Solutions
A Transition to Advanced Mathematics
- Let 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_forwardFind the order of each of the following elements in the multiplicative group of units . for for for forarrow_forwardFind two groups of order 6 that are not isomorphic.arrow_forward
- Find the right regular representation of G as defined Exercise 11 for each of the following groups. a. G={ 1,i,1,i } from Example 1. b. The octic group D4={ e,,2,3,,,, }.arrow_forward38. Let be the set of all matrices in that have the form with all three numbers , , and nonzero. Prove or disprove that is a group with respect to multiplication.arrow_forwardProve that Ca=Ca1, where Ca is the centralizer of a in the group G.arrow_forward
- If a is an element of order m in a group G and ak=e, prove that m divides k.arrow_forwardIn Exercises 15 and 16, the given table defines an operation of multiplication on the set S={ e,a,b,c }. In each case, find a condition in Definition 3.1 that fails to hold, and thereby show that S is not a group. See Figure 3.7 e a b c e e a b c a e a b c b e a b c c e a b carrow_forwardIn Exercises and, the given table defines an operation of multiplication on the set. In each case, find a condition in Definition that fails to hold, and thereby show that is not a group. 15. See Figure.arrow_forward
- 39. Let be the set of all matrices in that have the form for arbitrary real numbers , , and . Prove or disprove that is a group with respect to multiplication.arrow_forwardIf G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.arrow_forward12. 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_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning