Prove that the set of equivalence classes generated by ~ is uncountable

I am very unsure on how to do this question though I have attempted it. I'm not sure that the statement "if there exists a bijection between a set A and an uncountable set B, then A is uncountable" is true. I would appreciate a few pointers or some guidance on this question. Thanks

Let $T$ denote the set of real transcendental numbers.

Define $\sim$ to be an (equivalence) relation on $T$ defined by $x - y\in\Bbb{Q}$.

Let $C$ denote the set of equivalence classes generated by $\sim$.

Prove that the set $C$ is uncountable.

PROOF:

Note, the set of real numbers is uncountably infinite. Also note, the set of algebraic numbers (deonoted by $\Bbb{A}$) is countably infinite,

We have that $T=\Bbb{R}-\Bbb{A}$.

The set theoretic difference between between an uncountable and a countable set is uncountable. Hence, $T$ is uncountable.

Next, note that $\forall t\in T, \exists$ an equivalence class $[t]$.

Let the bijection $\sigma: T\to C$ be defined by $\sigma(t)=[t]$.

Hence, $C$ is uncountable.

Solution 1:

Your proof is wrong because $\sigma$ is not injective: $\sigma(e)=\sigma(e+1)$. Here is a correct proof. Note that every equivalence class $[a]$ is countable (it is in one-to-one correspondence with $\Bbb Q$: $q\mapsto a+q$). If the set of equivalence classes $C$ was countable, then $\Bbb R$ would be countable as the union of countably many countable sets.