Determine sets A and B

Question

I need help solving this task, if anyone had a similar problem it would help me.

The task is: Determine sets A and B if valid: $ A\cup B = \left\{x\in \mathbb{N} : x\le6 \right\}, A\cap B = \left\{x\in \mathbb{N} : x\le4 \right\}, \left\{4,6\right\} \not\subseteq A, \left\{5,6\right\} \not\subseteq B \setminus A.$

Thanks in advance!

Answer 1

So $\{1,2,3,4\} = A \cap B$ and $\{1,2,3,4, 5, 6\} = A \cup B$. Therefore $(A \cup B) \setminus (A \cap B) = \{5,6\}$. From $ \left\{4,6\right\} \not\subseteq A$, we deduce that $6 \notin A$ and $6 \in B$. And from $\left\{5,6\right\} \not\subseteq B \setminus A$ that $5 \in A$.

Finally $A = \{1,2,3,4, 5\}$ and $B = \{1,2,3,4, 6\}$.