Show that the set of letters needed to spell "$CATARACT$" and the set of letters needed to spell "$TRACT$" are equal.

In the following,state whether $A = B$ or not:
$A = \{2, 4, 6, 8, 10\}$; $B = \{x : x \text{ is a positive even integer and } x \le 10\}$

In each of the following,determine whether the statement is true or false. If it is true,prove it. If it is false,give an example.
If $A \subset B$ and $x \notin B,$ then $x \notin A$.

In a city,no two persons have an identical set of teeth,and there is no person without a tooth. Also,no person has more than $32$ teeth. If we disregard the shape and size of the teeth and consider only the presence or absence of a tooth at each of the $32$ positions,then the maximum population of the city is:

Consider the sets $\phi, A = \{1, 3\}, B = \{1, 5, 9\}, C = \{1, 3, 5, 7, 9\}$. Insert the symbol $\subset$ or $\not\subset$ between the following pair of sets: $\phi \dots B$.

State whether the following statement is true or false. Justify your answer.
${2, 6, 10}$ and ${3, 7, 11}$ are disjoint sets.