13b Real Mathematical Analysis Please provide complete and c

1.3.b. Real (Mathematical) Analysis. Please provide complete and correct solution done on computer or by hand with mathematical proof/explanation to all questions. Please, emphasis on complete and correct solution. The answers will be verified. This is all the information provided. Thank you very much.

3. In the following examples, show that the sets A and B are numerically equivalent by finding a specific bijection between the two (a) A 10, 1), B 10, 20) (b) A 10, 1), B 10, 1)

Solution

(a)

The following map is a bijection from A to B

f(x)=10x+10

(b)

Consider the sequence:

Xa=0,1,1/2,1/3,1/4,.....,1/n,..... in A

Label them as:xa1=0,xa2=1,xa3=1/2=1/(3-1),...xan=1/(n-1)

And

Xb=0,1/2,1/3,1/4,.....,1/n,..... in B

xb1=0,xb2=1/2,..., xbn=1/n,.....

So there is a bijection ,g from Xa to Xb

g(xan)=g(xbn)

And the sets:

A-Xa=B-Xb

so there is the trivial bijection from A-Xa to B-Xb which is the identity map

So the required bijection is

f(x)=x ,if x is in A-Xa

f(x)=g(x) if x is in Xa