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: $A \dots C$.

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:

Which of the following pairs of sets are equal? Justify your answer.
$X$,the set of letters in "$ALLOY$" and $B$,the set of letters in "$LOYAL$".

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

Let $x_{1}, x_{2}, \ldots, x_{15}$ be $15$ distinct numbers chosen from $1, 2, 3, \ldots, 15$. Then, the value of $(x_{1}-1)(x_{2}-1)(x_{3}-1) \ldots (x_{15}-1)$ is

Which of the following is a set? Justify your answer.
The collection of questions in this chapter.