Why is it that $3\in\{1, 2, 3\}$ but $\{1\}\notin \{1, 2\}$?

Question

Fill in the gap with either $\in$ or $\notin$

b) $\{1\}\;\_\_\; \{1, 2\}$. Answer says $\notin$.

but the other exercise a) $3\;\_\_\;\{1, 2, 3\}$, says that the answer is $\in$.

Why is it not $\in$ on b)? Is the answer in the book wrong here?

Also, what is the difference between "belong to" and a subset?

Answer 1

The elements of $\lbrace 1, 2 \rbrace$ are $1$ and $2$. Because $\lbrace 1 \rbrace \neq 1$ and $\lbrace 1 \rbrace \neq 2$, then $\lbrace 1 \rbrace$ is not an element of the set $\lbrace 1, 2 \rbrace$. So $\lbrace 1 \rbrace \notin \lbrace 1, 2 \rbrace$.

But the elements of $\lbrace 1, 2, 3 \rbrace$ are $1$, $2$ and $3$, so you see that $3$ is an element of the set $\lbrace 1, 2, 3 \rbrace$. So $3 \in \lbrace 1, 2, 3 \rbrace$.