Question

Prove that Q is infinite.

Please answer questions in clear hand-writing and show me the full process, thank you so much (Sometimes I get the answer which was difficult to read).

Answer 1

• Observe that the set of rational numbers is defined by:

(1)

Q={ab:a,b∈Z,b≠0}

• In fact, every rational number r can be uniquely written in the form r=pq where p,q∈Z, q≠0, and p and q are relatively prime, that is, the greatest common divisor of p and q is 1. For each rational number r∈Q we define a function f:Q→Z×Z by:

(2)

f(r)=(p,q)

• Then by the observation made above, f is injective. Furthermore, since Z is countable we have that Z×Z is countable by the theorem presented on The Cartesian Product of Two Countable Sets is Countable . Therefore f(Q) is countably infinite and so there exists a bijection g:f(Q)→N.

• Note that the function h:Q→f(Q) defined by h(q)=f(q) is a bijection. Therefore the composition g∘h:Q→N is a bijection. So Q is countably infinite. Hence Q is infinite.■