Suppose that A = {2, 4, 6}, B = {2, 6}, C = {4, 6}, and D = {4, 6, 8}. Determine which of these sets are subsets of which other of these sets.

Suppose that $A = \{2, 4, 6\}, B = \{2, 6\}, C = \{4, 6\},$ and $D = \{4, 6, 8\}.$ Let us determine which of these sets are subsets of which other of these sets. 

It follows that $B\subset A$ because of $2\in A$ and $6\in A.$ Since $2\notin C$ and $2\notin D,$ $B$ is not a subset of $C$ and $B$ is not a subset of $D.$ Taking into account that $4\in A$ and $6\in A,$ we conclude that $C\subset A.$ By analogy, since $4\in D$ and $6\in D,$ we conclude that $C\subset D.$ Since $|A|>|B|$ and $|A|>|C|,$ we conclude that $A$ is not a subset of $B$ and $A$ is not a subset of $C.$ Taking into account that $|D|>|B|$ and $|D|>|C|,$ we conclude that $D$ is not a subset of $B$ and $D$ is not a subset of $C.$ Since $2\notin D,$ $A$ is not a subset of $D.$ Since $8\notin A,$ $D$ is not a subset of $A.$