Let $Z$ be the set of all integers,$A = \{(x, y) \in Z \times Z : (x-2)^{2} + y^{2} \leq 4\}$,$B = \{(x, y) \in Z \times Z : x^{2} + y^{2} \leq 4\}$,and $C = \{(x, y) \in Z \times Z : (x-2)^{2} + (y-2)^{2} \leq 4\}$. If the total number of relations from $A \cap B$ to $A \cap C$ is $2^{p}$,then the value of $p$ is:

On the set of all real numbers,a relation $R$ is defined as $a \, R \, b$ if and only if $|a - b| \le 1$. Then $R$ is:

The relation $R$ defined on a set $A$ is antisymmetric if $(a, b) \in R$ and $(b, a) \in R$ implies $a = b$ for all $a, b \in A$. Based on this definition,the relation $R$ is antisymmetric if $(a, b) \in R$ and $(b, a) \in R$ implies $a = b$,which is equivalent to saying that if $a \neq b$,then it is not possible for both $(a, b) \in R$ and $(b, a) \in R$ to be true. Therefore,the condition is that for $a \neq b$,we cannot have both $(a, b) \in R$ and $(b, a) \in R$.

On the set $R$ of real numbers, we define $x P y$ if and only if $x y \geq 0$. Then, the relation $P$ is

Let $R$ be a relation on the set $A$ of ordered pairs of positive integers defined by $(x, y) R (u, v)$ if and only if $xv = yu$. Show that $R$ is an equivalence relation.

Let $L$ be the set of all straight lines in the Euclidean plane. Two lines $l_1$ and $l_2$ are said to be related by the relation $R$ if and only if $l_1$ is parallel to $l_2$. Then the relation $R$ is