Using the standard diangonal argument which shows that the s

Using the standard \"diangonal\" argument which shows that the set of positive rational numbers is denumerable, what is the 7-th rational number. Please explain your answer.

Solution

A set is denumerable if it can put in to one to one correspondence with natural number

first we arreng the positive rational number in sequence

i. let {r₁,r_2,r₃----------}

we construct the doubly infinit arrey

i.e

1/1   2/1       3/1   4/1 5/1------------                   

1/2    2/2       3/2   4/2 5/2---------------

1/3    2/3      3/3   4/3 5/3-------------

1/4      2/4      3/4   4/4 5/4-----------

which contains evry positive rational number

so ther is a one to one correspondence with natural numaber

therefore

By defination of denumerable set

positive rational number is denumerable

now order these numbersby starting at the top left hand cornar and going up and dawn the digonals sequentially

omitting the terms which have already been counted

consider

1/1   2/1       3/1   4/1 5/1------------                   

1/2    2/2       3/2   4/2 5/2---------------

1/3    2/3      3/3   4/3 5/3-------------

1/4      2/4      3/4   4/4 5/4-----------

therefore we get 1,2,1/2,1/3,3,4,3/2,2/3,1/4,------

therefor 7 th rational number is 3/2