The number of relations $R$ from an $m$-element set $A$ to an $n$-element set $B$ satisfying the condition $(a, b_1) \in R, (a, b_2) \in R \Rightarrow b_1 = b_2$ for $a \in A, b_1, b_2 \in B$ is

Statement $1$ : If $A$ and $B$ are two sets having $p$ and $q$ elements respectively,where $q > p$. Then the total number of functions from set $A$ to set $B$ is $q^p$.
Statement $2$ : The total number of selections of $p$ different objects out of $q$ objects is ${}^qC_p$.

The function $t$ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by $t(C) = \frac{9C}{5} + 32$. Find $t(-10)$.

Let $N$ be the set of natural numbers and the relation $R$ be defined on $N$ such that $R = \{(x, y) : y = 2x, x, y \in N \}$. What is the domain,codomain,and range of $R$? Is this relation a function?

Let $f$ be the subset of $Z \times Z$ defined by $f = \{(ab, a+b) : a, b \in Z\}$. Is $f$ a function from $Z$ to $Z$? Justify your answer.

Let $f: X \rightarrow Y$ be a function and $A_y = f^{-1}(\{y\})$ for $y \in Y$. Then $A_i \cap A_j = \phi$ $(i \neq j)$ for all $i, j \in Y$ and $\bigcup_{y \in Y} A_y = X$, if