السلام عليكم ورحمة الله وبركاته
لاثبات ان N , N_2 متكافئان علينا ايجاد دالة او تقابل بين N , N_2 بحيث تكون هذه الدالة one to one and onto
f(x)=x^2
such that f:N_______>N^2
1- we prove that f is one to one mapping
let f(X)=f(y), so x^2=y^2< then x=y because x, y are natural numer
then f is one to one

2- we prove that is onto
let y^2 an element in N^2, so the squar root of y^2 equal y, then for all y^2 in N^2 there is y such that f(y)=y^2

bye