1. Determine the sets $A$ and $B$ for which the following five properties hold:

- $A \cup B=\{1,2,3,4,5,6\}$;
- $2 \in A \backslash B$;
- $3 \in B \backslash A$;
- $A \cap\{4,5,6\}=\varnothing$;
- $B \cap\{1\}=\varnothing$.
1. From the fourth condition (along with the first), we get that the set $A$ can only contain the numbers 1, 2, and 3. From the third condition, we have $3 \notin A$, and from the second, $2 \in A$. Further, from the last condition, we have $1 \notin B$, and since $1 \in A \cup B$, it follows that $1 \in A$. Thus, we have uniquely identified the set $A: A=\{1,2\}$. The set $B$ must contain all the numbers $3,4,5$ and 6, to satisfy the first condition. From the second condition, we also have $2 \notin B$, and from the last, $1 \notin B$. Therefore, the only possibility is $B=\{3,4,5,6\}$.