Question

Consider the sets A and B, where A = {3, |B|} and B = {1, |A|, |B|}. What are the sets?

Answer 1

Consider the sets A and B, where A = {3, |B|} and B = {1, |A|, |B|}. 

Let us prove by contradiction that $|B|\ne 3.$ Suppose that |B| = 3. Then cardinality of the set A = {3, 3} = {3} is equal to 1, and hence the set B = {1, |A|, |B|} = {1, 1, 3} = {1, 3} has cardilality 2. And we have a contradiction with |B| = 3.

Therefore, $|B|\ne 3,$ and hence $|B|<3.$ It follows that $|A|=2.$ Then $B = \{1, |A|, |B|\}=\{1, 2,|B|\},$ and hence $|B|=2.$

We conclude that $A=\{3,2\}$ and $B=\{1,2\}.$