Question

(a) Recall that two sets have the same cardinality if there is a bijection between them and that Z is the set of all integers. Give an example of a bijection f: Z+Z which is different from the identity function. (b) For the following sets A prove that A has the same cardinality as the positive integers Z+ i. A= {r eZ+By Z r = y²} ii. A=Z 1.

Answer 1

ile la) Let set of ila integers. 2lo اد داد if ther. use ola Z is the all Example! function, f: ZZ define by f(n) = n+1 for all nez Here, f is bijective function. b) Definition: Two sets have the same cardinality between is bijective them. I will This fact. A={xe7+ | 3 YEZ x=12} define the function, f: 7 = f(n) = 12, for no M₂ Ezt a nezt

e So, fim) - fim,) =) n = n - n = n₂ So f is one to one. for there exist NEA nEzt St. any frm=n=m2 (nm2€ A) f is onto. Hence, f is bijective. has same cardinalivy the zt, A - Z define the function, f: Z + A, g(n) = 1 it n is even 1 if n is odd So, as positive integer C 2 1-n 2

dre eren for M, N, EZ+ n, and n₂ geni)= gen) Ол 2 =) n = n₂ and for odd m, m₂ are 0 m, m₂ € 2 + g(mi)= gema) =)1-m = 1m2 110 =) 1- 1 1 1 2 =) m = m₂ So, to one ome g is for any 105 ht XEA is a possitive Integer. 8120)= possitive integer J(1-2* and if is 112ilo non- 2x)= x so g is onto So, g Is bijective. as cardinality the positive Hence, A same has integer et.