One-to-one correspondence of a set within a set

Question

I need to find a one-to-one correspondence between each of the following pairs of sets:

1. $\{x, y, \{a, b, c\}\}$ and $\{14, -3, t\}$
2. $2\mathbb Z$ and $17 \mathbb Z$

For problem a, I have no idea if the inner set counts as a single element and therefore am not sure. For problem b, I'm guessing it does have a one-to-one correspondence simply such that $1 \mapsto 17$, $2 \mapsto 34$, etc.

Any help is greatly appreciated.

EDIT: Is this more basic than I thought? I seem to be getting downvoted a fair amount. Just trying to improve my math skills...

Answer 1

For (a) we can map $x \mapsto 14$, $y \mapsto -3$ and $\{a, b, c\} \mapsto t$. So to answer your question, $\{a, b, c\}$ is considered one element.

For (b) note that $2 \mathbb Z = \{2n : n \in \mathbb Z \}$ and $7 \mathbb Z = \{7n : n \in \mathbb Z\}$. So for any $x \in 2\mathbb Z$, $x = 2n$ for some $n \in \mathbb Z$. Map it via $x \mapsto 7n$ since $n$ uniquely identifies $x$.

Answer 2

Indeed $\{a,b,c\}\in \{x,y,\{a,b,c\}\}$, i.e. it is an element. In fact, if this weren't the case, it would be impossible to find a bijection because the two sets would have an unequal number of elements.

I think you already have the right idea for the second part of the question.