Prove that the sets E = {x ∈ ℕ : x = 2k, k ∈ ℕ } and ℕ have the same cardinality.
A clear definition of cardinality was not given in this situation, so I understand cardinality to be (more or less) the equivalence of the elements in any two given sets. So how would I go about proving the cardinality of these two sets to be the same?
We say that two set has same cardinality if there is a one to one correspondence between the two sets. The set $E$ in your question is the set of all Even numbers which is a subset of the natural number set. We can define a function as
$f:\mathbb{N} \to E$ as $f(x)=2x$ where $x\in \mathbb{N}$. So now you have to prove that this function is bijective and done!