State whether the following set is finite or infinite:
The set of circles passing through the origin $(0,0)$.

Let $A, B$ and $C$ be three sets. If $A \in B$ and $B \subset C$,is it true that $A \subset C$? If not,give an example.

Two sets $A$ and $B$ are disjoint if and only if

Let $A$ and $B$ be sets. If $A \cap X = B \cap X = \phi$ and $A \cup X = B \cup X$ for some set $X$,show that $A = B$. (Hint: Use $A = A \cap (A \cup X)$,$B = B \cap (B \cup X)$ and the Distributive law.)

The remainder obtained when the polynomial $1 + x + x^3 + x^9 + x^{27} + x^{81} + x^{243}$ is divided by $x - 1$ is

Let $S = \{1, 2, 3, \ldots, n\}$ and $A = \{(a, b) \mid 1 \leq a, b \leq n\} = S \times S$. $A$ subset $B$ of $A$ is said to be a good subset if $(x, x) \in B$ for every $x \in S$. Then,the number of good subsets of $A$ is