real analysis - Which of the following is uncountable?

Which of the following is uncountable?

$1.\{f|f:\{0,1\}\to\mathbb{Z}\}$

$2.\{f|f:\mathbb{Z}\to\{0,1\}\}$

My attempt:I think first is countable because we can make a bijection from this to $\mathbb{Z^2}$,am I correct?About second option I do not have any idea.

Thanks.

Answer

Hint for 2: $\{f\mid f:\mathbb{Z}\to\{0,1\}\}$ has the same cardinality as the power set of $\mathbb{Z}$.