The number of proper subsets of the set $\{1, 2, 3\}$ is

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:

Decide,among the following sets,which sets are subsets of one another:
$A = \{ x: x \in \mathbb{R} \text{ and } x \text{ satisfies } x^2 - 8x + 12 = 0 \},$
$B = \{2, 4, 6\}, C = \{2, 4, 6, 8, \dots\}, D = \{6\}$

If $aN = \{ax : x \in N\}$ and $bN \cap cN = dN$,where $b, c \in N$ are coprime numbers,then:

The dual of $(x+y) \cdot (x^{\prime} \cdot 1)$ is:

If $P, Q$ and $R$ are subsets of $A$,then $R \times (P \cup Q) = $