In set-builder form,the empty set is represented by

In this question,all integers are represented in base $10$. Consider the set $E$ of positive integers $n$ having the property that when any nonzero digit $d \in \{1, 2, \dots, 9\}$ is written to the right of $n$,the resulting number is divisible by $d$. Let $N$ be the smallest element of $E$. The product of the digits of $N$ is:

Make correct statements by filling in the symbols $\subset$ or $\not\subset$ in the blank spaces:
${ x:x \text{ is a circle in the plane} } \dots { x:x \text{ is a circle in the same plane with radius } 1 \text{ unit} }$

Show that the following four conditions are equivalent:
$(i) A \subset B, \quad (ii) A - B = \phi, \quad (iii) A \cup B = B, \quad (iv) A \cap B = A$

Given the sets $A = \{1, 3, 5\}$,$B = \{2, 4, 6\}$,and $C = \{0, 2, 4, 6, 8\}$,which of the following sets can be considered as a universal set for all three sets $A$,$B$,and $C$?

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